Selina Concise Solutions for ICSE Class 7 Mathematics Chapter 11 Fundamental Concepts Including Fundamental Operations

Points to Remember

• Constants and Variables: A constant is a number with a fixed value. A variable is represented by a letter from the English alphabet that can be assigned different values depending on the mathematical requirements.
• Term: A term refers to a single number (constant), a variable, or any multiplication or division of numbers and variables.
• Algebraic Expression: This is a collection of one or more terms connected together by addition (+) or subtraction (-) signs.
• Types of Algebraic Expressions:
  • (i) Monomial: Contains only a single term.
  • (ii) Binomial: Contains two distinct terms.
  • (iii) Trinomial: Contains three distinct terms.
  • (iv) Multinomial: Contains more than three terms.
  • (v) Polynomial: Contains two or more terms.
  Note: An expression like \( \frac{2}{x} \) is not considered a monomial unless \( x \) is a non-zero value.
• Product: When two or more quantities are multiplied together, the resulting value is called their product.
• Factors: Each of the quantities (either numbers or variables) multiplied together to form a term is called a factor of that term.
• Co-efficient: Within a monomial, any factor or group of factors is referred to as the co-efficient of the remaining part of that term.
• Degree of a Monomial: The degree of a monomial is the exponent of its variable, or the sum of the exponents of all its variables.
• Degree of a Polynomial: The degree of a polynomial is determined by the term with the highest degree in that expression.
• Like and Unlike Terms: Terms that share the exact same literal coefficients or variables are called like terms. Terms with different variables or literal coefficients are unlike terms.
• Addition and Subtraction: You can only add or subtract like terms by combining their numerical coefficients.
• Multiplication and Division:
  • (A) Multiplication:
    • (i) Multiplying monomials:
    • (a) Multiply the numerical coefficients together.
    • (b) Multiply the literal coefficients separately.
    • (c) Combine any like terms together.
  • (B) Division:
    • (i) To divide a polynomial by a monomial, divide each term of the polynomial by that monomial and simplify each resulting fraction.
    • (ii) To divide one polynomial by another polynomial, first arrange the terms of both expressions in descending or ascending order of their powers, and then proceed with the division.

Some Important Points

Types of Brackets:

The names of different brackets and the standard order in which they are removed are as follows:

• (a) Bar (Vinculum) bracket: \( \overline{\quad} \)
• (b) Circular bracket: ( )
• (c) Curly bracket: { }
• (d) Square bracket: [ ]

Exercise 11(A)

Question 1. Separate constant terms and variable terms from the following:
\( 8, x, 6xy, 6 + x, -5xy^2, 15az^2, \frac{32z}{xy}, \frac{y^2}{3x} \)
Answer: In this list, 8 is the only constant. All of the other terms are variables.
In simple words: The number 8 has a fixed value, so it is a constant. The others have letters, so they are variables.

Exam Tip: Always remember that a constant has a fixed numerical value like 8, whereas any term with a letter in it is a variable because its value can change.

Question 2. Constant is only 8 others are variables
(i) \( 2x \div 15 \)
(ii) \( ax + 9 \)
(iii) \( 3x^2 \times 5x \)
(iv) \( 5 + 2a - 3b \)
(v) \( 2y - \frac{7}{3}z \div x \)
(vi) \( 3p \times q \div z \)
(vii) \( 12z \div 5x + 4 \)
(viii) \( 12 - 5z - 4 \)
(ix) \( a^3 - 3ab^2 \times c \)
Answer:
(i) \( 2x \div 15 = \frac{2x}{15} \): This is a monomial because it consists of a single term.
(ii) \( ax + 9 \): It is a binomial since it contains two terms.
(iii) \( 3x^2 \times 5x = 15x^3 \): It is a monomial because it simplifies to a single term.
(iv) \( 5 + 2a - 3b \): It is a trinomial since there are three terms.
(v) \( 2y - \frac{7}{3}z \div x = 2y - \frac{7z}{3x} \): This represents a binomial as it consists of two terms.
(vi) \( 3p \times q \div z = \frac{3pq}{z} \): This is a monomial because it has only one term.
(vii) \( 12z \div 5x + 4 = \frac{12z}{5x} + 4 \): This is a binomial because it contains two terms.
(viii) \( 12 - 5z - 4 = 8 - 5z \): This is a binomial since it reduces to two terms.
(ix) \( a^3 - 3ab^2 \times c = a^3 - 3ab^2c \): This is a binomial since it contains two terms.
In simple words: Expressions with only one term are monomials, those with two terms are binomials, and those with three terms are trinomials. Always simplify multiplication and division first to see the real terms.

Exam Tip: Never count the terms before simplifying. Operations like multiplication and division combine numbers into a single term, while only plus and minus signs separate different terms.

Question 3. Write the coefficient of:
(i) \( xy \) in \( -3axy \)
(ii) \( z^2 \) in \( p^2yz^2 \)
(iii) \( mn \) in \( -mn \)
(iv) \( 15 \) in \( -15p^2 \)
Answer:
(i) The coefficient of \( xy \) in \( -3axy \) is \( -3a \).
(ii) The coefficient of \( z^2 \) in \( p^2yz^2 \) is \( p^2y \).
(iii) The coefficient of \( mn \) in \( -mn \) is \( -1 \).
(iv) The coefficient of \( 15 \) in \( -15p^2 \) is \( -p^2 \).
In simple words: To find the coefficient of a part, just remove that part from the term and see what is left over.

Exam Tip: Remember to include the negative sign in your coefficient if the original term is negative, and if nothing else is left, the coefficient is 1 or -1.

Question 4. For each of the following monomials, write its degree:
(i) \( 7y \)
(ii) \( -x^2y \)
(iii) \( xy^2z \)
(iv) \( -9y^2z^3 \)
(v) \( 3m^3n^4 \)
(vi) \( -2p^2q^3r^4 \)
Answer:
(i) The degree of \( 7y \) is \( 1 \).
(ii) The degree of \( -x^2y \) is \( 2 + 1 = 3 \).
(iii) The degree of \( xy^2z \) is \( 1 + 2 + 1 = 4 \).
(iv) The degree of \( -9y^2z^3 \) is \( 2 + 3 = 5 \).
(v) The degree of \( 3m^3n^4 \) is \( 3 + 4 = 7 \).
(vi) The degree of \( -2p^2q^3r^4 \) is \( 2 + 3 + 4 = 9 \).
In simple words: To find the degree of a single term, add up the powers of all the letters in it. If a letter has no power shown, its power is 1.

Exam Tip: Don't forget that a variable with no written exponent, like x or y, has an exponent of 1. Always count it when adding exponents to find the degree of a monomial.

Question 5. Write the degree of each of the following polynomials:
(i) \( 3y^3 - x^2y^2 + 4x \)
(ii) \( p^3q^2 - 6p^2q^5 + p^4q^4 \)
(iii) \( -8mn^6 + 5m^3n \)
(iv) \( 7 - 3x^2y + y^2 \)
(v) \( 3x - 15 \)
(vi) \( 2y^2z + 9yz^3 \)
Answer:
(i) The degree of \( 3y^3 - x^2y^2 + 4x \) is \( 4 \), because \( -x^2y^2 \) has the highest degree (2 + 2 = 4).
(ii) The degree of \( p^3q^2 - 6p^2q^5 + p^4q^4 \) is \( 8 \), as the term \( p^4q^4 \) has the highest degree (4 + 4 = 8).
(iii) The degree of \( -8mn^6 + 5m^3n \) is \( 7 \), because the term \( -8mn^6 \) has the highest degree (1 + 6 = 7).
(iv) The degree of \( 7 - 3x^2y + y^2 \) is \( 3 \), as \( -3x^2y \) is the highest degree term (2 + 1 = 3).
(v) The degree of \( 3x - 15 \) is \( 1 \), because \( 3x \) is the term with the highest degree.
(vi) The degree of \( 2y^2z + 9yz^3 \) is \( 4 \), as \( 9yz^3 \) is the highest degree term (1 + 3 = 4).
In simple words: To find the degree of a polynomial, check the degree of each term separately. The biggest degree you find is the degree of the whole polynomial.

Exam Tip: Find the sum of the exponents of variables in each individual term first. The maximum sum among all terms is the degree of the polynomial.

Question 6. Group the like term together:
(i) \( 9x^2 \), \( xy \), \( -3x^2 \), \( x^2 \) and \( -2xy \)
(ii) \( ab \), \( -a^2b \), \( -3ab \), \( 5a^2b \) and \( -8a^2b \)
(iii) \( 7p \), \( 8pq \), \( -5pq \), \( -2p \) and \( 3p \)
Answer:
(i) The like terms are \( 9x^2, -3x^2, x^2 \) in one group, and \( xy, -2xy \) in another group.
(ii) The groups of like terms are \( ab, -3ab \) and \( -a^2b, 5a^2b, -8a^2b \).
(iii) The like terms are \( 7p, -2p, 3p \) as one set, and \( 8pq, -5pq \) as another set.
In simple words: Like terms are terms that have exactly the same letters raised to the same powers. We can group them together.

Exam Tip: When identifying like terms, pay close attention to the powers of the variables. For example, ab and a^2b are unlike terms because the power of 'a' is different.

Question 7. Write numerical co-efficient of each of the followings:
(i) \( y \)
(ii) \( -y \)
(iii) \( 2x^2y \)
(iv) \( -8xy^3 \)
(v) \( 3py^2 \)
(vi) \( -9a^2b^3 \)
Answer:
(i) The numerical coefficient of \( y \) is \( 1 \).
(ii) The numerical coefficient of \( -y \) is \( -1 \).
(iii) The numerical coefficient of \( 2x^2y \) is \( 2 \).
(iv) The numerical coefficient of \( -8xy^3 \) is \( -8 \).
(v) The numerical coefficient of \( 3py^2 \) is \( 3 \).
(vi) The numerical coefficient of \( -9a^2b^3 \) is \( -9 \).
In simple words: The numerical coefficient is simply the number multiplying the variables in a term. If there is only a minus sign, it is -1, and if there is no number, it is 1.

Exam Tip: Don't forget that a term with a negative sign but no number, like -y, has a numerical coefficient of -1.

Question 8. In \( -5x^3y^2z^4 \); write the coefficient of:
(i) \( z^2 \)
(ii) \( y^2 \)
(iii) \( yz^2 \)
(iv) \( x^3y \)
(v) \( -xy^2 \)
(vi) \( -5xy^2z \)
Also, write the degree of the given algebraic expression.
Answer:
(i) The coefficient of \( z^2 \) in \( -5x^3y^2z^4 \) is \( -5x^3y^2z^2 \).
(ii) The coefficient of \( y^2 \) in \( -5x^3y^2z^4 \) is \( -5x^3z^4 \).
(iii) The coefficient of \( yz^2 \) in \( -5x^3y^2z^4 \) is \( -5x^3yz^2 \).
(iv) The coefficient of \( x^3y \) in \( -5x^3y^2z^4 \) is \( -5yz^4 \).
(v) The coefficient of \( -xy^2 \) in \( -5x^3y^2z^4 \) is \( 5x^2z^4 \).
(vi) The coefficient of \( -5xy^2z \) in \( -5x^3y^2z^4 \) is \( x^2z^3 \).
The total degree of this algebraic term is \( 3 + 2 + 4 = 9 \).
In simple words: To find the coefficient of any part, divide the whole term by that part. The remaining quotient is your coefficient. To find the degree, add up all the powers.

Exam Tip: Be very careful with signs when finding coefficients of negative parts. For example, dividing a negative term by a negative part results in a positive coefficient.

Exercise 11(B)

Question 1. Fill in the blanks:
(i) \( 8x + 5x = \dots \dots \)
(ii) \( 8x - 5x = \dots \dots \)
(iii) \( 6xy^2 + 9xy^2 = \dots \dots \)
(iv) \( 6xy^2 - 9xy^2 = \dots \dots \)
(v) The sum of \( 8a \), \( 6a \) and \( 5b = \dots \dots \)
(vi) The addition of \( 5 \), \( 7xy \), \( 6 \) and \( 3xy = \dots \dots \)
(vii) \( 4a + 3b - 7a + 4b = \dots \dots \)
(viii) \( -15x + 13x + 8 = \dots \dots \)
(ix) \( 6x^2y + 13xy^2 - 4x^2y + 2xy^2 = \dots \dots \)
(x) \( 16x^2 - 9x^2 = \dots \dots \) and \( 25xy^2 - 17xy^2 = \dots \dots \)
Answer:
(i) \( 8x + 5x = 13x \)
(ii) \( 8x - 5x = 3x \)
(iii) \( 6xy^2 + 9xy^2 = 15xy^2 \)
(iv) \( 6xy^2 - 9xy^2 = -3xy^2 \)
(v) The sum of \( 8a \), \( 6a \) and \( 5b \) is \( 8a + 6a + 5b = 14a + 5b \)
(vi) Adding \( 5 \), \( 7xy \), \( 6 \) and \( 3xy \) gives \( 5 + 6 + 7xy + 3xy = 11 + 10xy \)
(vii) \( 4a + 3b - 7a + 4b = 4a - 7a + 3b + 4b = -3a + 7b = 7b - 3a \)
(viii) \( -15x + 13x + 8 = -2x + 8 = 8 - 2x \)
(ix) \( 6x^2y + 13xy^2 - 4x^2y + 2xy^2 = (6x^2y - 4x^2y) + (13xy^2 + 2xy^2) = 2x^2y + 15xy^2 \)
(x) \( 16x^2 - 9x^2 = 7x^2 \) and \( 25xy^2 - 17xy^2 = 8xy^2 \)
In simple words: Only combine like terms together. You cannot add or subtract terms with different letters or different powers.

Exam Tip: Be careful to group like terms with their correct signs before adding or subtracting their numerical coefficients.

Question 2. Add:
(i) \( -9x \), \( 3x \) and \( 4x \)
(ii) \( 23y^2 \), \( 8y^2 \) and \( -12y^2 \)
(iii) \( 18pq - 15pq \) and \( 3pq \)
Answer:
(i) \( -9x + 3x + 4x = -9x + 7x = -2x \)
(ii) \( 23y^2 + 8y^2 + (-12y^2) = 31y^2 - 12y^2 = 19y^2 \)
(iii) \( 18pq - 15pq + 3pq = (18pq + 3pq) - 15pq = 21pq - 15pq = 6pq \)
In simple words: To add like terms, just add their numerical coefficients while keeping the variables exactly the same.

Exam Tip: Add positive terms first to make calculation simpler and reduce the chance of sign errors.

Question 3. Simplify:
(i) \( 3m + 12m - 5m \)
(ii) \( 7n^2 - 9n^2 + 3n^2 \)
(iii) \( 25zy - 8zy - 6zy \)
(iv) \( -5ax^2 + 7ax^2 - 12ax^2 \)
(v) \( -16am + 4mx + 4am - 15mx + 5am \)
Answer:
(i) \( 3m + 12m - 5m = 15m - 5m = 10m \)
(ii) \( 7n^2 - 9n^2 + 3n^2 = (7n^2 + 3n^2) - 9n^2 = 10n^2 - 9n^2 = n^2 \)
(iii) \( 25zy - 8zy - 6zy = 25zy - (8zy + 6zy) = 25zy - 14zy = 11zy \)
(iv) \( -5ax^2 + 7ax^2 - 12ax^2 = (-5ax^2 - 12ax^2) + 7ax^2 = -17ax^2 + 7ax^2 = -10ax^2 \)
(v) \( -16am + 4mx + 4am - 15mx + 5am = (-16am + 4am + 5am) + (4mx - 15mx) = -7am - 11mx \)
In simple words: Combine all positive like terms first, then combine the negative ones, and finally subtract to get the final simplified term.

Exam Tip: Be careful with the coefficients of terms like n^2, where the coefficient of 1 is understood and usually not written.

Question 4. Add:
(i) \( a + b \) and \( 2a + 3b \)
(ii) \( 2x + y \) and \( 3x - 4y \)
(iii) \( -3a + 2b \) and \( 3a + b \)
(iv) \( 4 + x \), \( 5 - 2x \) and \( 6x \)
Answer:
(i) \( (a + b) + (2a + 3b) = a + 2a + b + 3b = 3a + 4b \)
(ii) \( (2x + y) + (3x - 4y) = (2x + 3x) + (y - 4y) = 5x - 3y \)
(iii) \( (-3a + 2b) + (3a + b) = (-3a + 3a) + (2b + b) = 0 + 3b = 3b \)
(iv) \( (4 + x) + (5 - 2x) + 6x = (x - 2x + 6x) + (4 + 5) = 5x + 9 \)
In simple words: To add two or more algebraic expressions, group the like terms together and add them, while keeping the constants separate.

Exam Tip: When terms with equal and opposite coefficients are added, like -3a and 3a, they cancel each other out to become 0.

Question 5. Find the sum of:
(i) \( 3x + 8y + 7z \), \( 6y + 4z - 2x \) and \( 3y - 4x + 6z \)
(ii) \( 3a + 5b + 2c \), \( 2a + 3b - c \) and \( a + b + c \)
(iii) \( 4x^2 + 8xy - 2y^2 \) and \( 8xy - 5y^2 + x^2 \)
(iv) \( 9x^2 - 6x + 7 \), \( 5 - 4x \) and \( 6 - 3x^2 \)
(v) \( 5x^2 - 2xy + 3y^2 \) and \( -2x^2 + 5xy + 9y^2 \) and \( 3x^2 - xy - 4y^2 \)
(vi) \( a^2 + b^2 + 2ab \), \( 2b^2 + c^2 + 2bc \) and \( 4c^2 - a^2 + 2ac \)
(vii) \( 9ax - 6bx + 8 \), \( 4ax + 8bx - 7 \) and \( -6ax - 4bx - 3 \)
(viii) \( abc + 2ba + 3ac \), \( 4ca - 4ab + 2bca \) and \( 2ab - 3abc - 6ac \)
(ix) \( 4a^2 + 5b^2 - 6ab \), \( 3ab \), \( 6a^2 - 2b^2 \) and \( 4b^2 - 5ab \)
(x) \( x^2 + x - 2 \), \( 2x - 3x^2 + 5 \) and \( 2x^2 - 5x + 7 \)
(xi) \( 4x^3 + 2x^2 - x + 1 \), \( 2x^3 - 5x^2 - 3x + 6 \), \( x^2 + 8 \) and \( 5x^3 - 7x \)
Answer:
(i) \( (3x + 8y + 7z) + (6y + 4z - 2x) + (3y - 4x + 6z) = (3x - 2x - 4x) + (8y + 6y + 3y) + (7z + 4z + 6z) = -3x + 17y + 17z \)
(ii) \( (3a + 5b + 2c) + (2a + 3b - c) + (a + b + c) = (3a + 2a + a) + (5b + 3b + b) + (2c - c + c) = 6a + 9b + 2c \)
(iii) \( (4x^2 + 8xy - 2y^2) + (8xy - 5y^2 + x^2) = (4x^2 + x^2) + (8xy + 8xy) + (-2y^2 - 5y^2) = 5x^2 + 16xy - 7y^2 \)
(iv) \( (9x^2 - 6x + 7) + (5 - 4x) + (6 - 3x^2) = (9x^2 - 3x^2) + (-6x - 4x) + (7 + 5 + 6) = 6x^2 - 10x + 18 \)
(v) \( (5x^2 - 2xy + 3y^2) + (-2x^2 + 5xy + 9y^2) + (3x^2 - xy - 4y^2) = (5x^2 - 2x^2 + 3x^2) + (-2xy + 5xy - xy) + (3y^2 + 9y^2 - 4y^2) = 6x^2 + 2xy + 8y^2 \)
(vi) \( (a^2 + b^2 + 2ab) + (2b^2 + c^2 + 2bc) + (4c^2 - a^2 + 2ac) = (a^2 - a^2) + (b^2 + 2b^2) + (c^2 + 4c^2) + 2ab + 2bc + 2ac = 3b^2 + 5c^2 + 2ab + 2bc + 2ac \)
(vii) \( (9ax - 6bx + 8) + (4ax + 8bx - 7) + (-6ax - 4bx - 3) = (9ax + 4ax - 6ax) + (-6bx + 8bx - 4bx) + (8 - 7 - 3) = 7ax - 2bx - 2 \)
(viii) \( (abc + 2ba + 3ac) + (4ca - 4ab + 2bca) + (2ab - 3abc - 6ac) = (abc + 2abc - 3abc) + (2ab - 4ab + 2ab) + (3ac + 4ac - 6ac) = 0 + 0 + ca = ca \)
(ix) \( (4a^2 + 5b^2 - 6ab) + 3ab + (6a^2 - 2b^2) + (4b^2 - 5ab) = (4a^2 + 6a^2) + (5b^2 - 2b^2 + 4b^2) + (-6ab + 3ab - 5ab) = 10a^2 + 7b^2 - 8ab \)
(x) \( (x^2 + x - 2) + (2x - 3x^2 + 5) + (2x^2 - 5x + 7) = (x^2 - 3x^2 + 2x^2) + (x + 2x - 5x) + (-2 + 5 + 7) = 0x^2 - 2x + 10 = -2x + 10 \)
(xi) \( (4x^3 + 2x^2 - x + 1) + (2x^3 - 5x^2 - 3x + 6) + (x^2 + 8) + (5x^3 - 7x) = (4x^3 + 2x^3 + 5x^3) + (2x^2 - 5x^2 + x^2) + (-x - 3x - 7x) + (1 + 6 + 8) = 11x^3 - 2x^2 - 11x + 15 \)
In simple words: To find the sum of multiple expressions, group all terms with the exact same variables and powers together, add their coefficients, and write down the simplified result.

Exam Tip: Be very careful when grouping terms with signs. If a term has a negative sign in front, make sure that negative sign stays with it when you move it.

Question 6. Find the sum of:
(i) \( x \) and \( 3y \)
(ii) \( -2a \) and \( +5 \)
(iii) \( -4x^2 \) and \( +7x \)
(iv) \( +4a \) and \( -7b \)
(v) \( x^3 + 3x^2y \) and \( 2y^2 \)
(vi) \( 11 \) and \( -by \)
Answer:
(i) The sum of \( x \) and \( 3y \) is \( x + 3y \).
(ii) The sum of \( -2a \) and \( +5 \) is \( -2a + 5 \).
(iii) The sum of \( -4x^2 \) and \( +7x \) is \( -4x^2 + 7x \).
(iv) The sum of \( +4a \) and \( -7b \) is \( 4a - 7b \).
(v) The sum of \( x^3 + 3x^2y \) and \( 2y^2 \) is \( x^3 + 3x^2y + 2y^2 \).
(vi) The sum of \( 11 \) and \( -by \) is \( 11 - by \).
In simple words: Since these terms are unlike, they cannot be simplified further. We just write them together connected by a plus or minus sign.

Exam Tip: Do not try to combine unlike terms (like x and 3y) into a single term. They must be left separate as an algebraic expression.

Question 7. The sides of a triangle are \( 2x + 3y \), \( x + 5y \) and \( 7x - 2y \), find its perimeter.
Answer: The three sides of the triangle are \( 2x + 3y \), \( x + 5y \), and \( 7x - 2y \).
The perimeter is the sum of these three sides:
\( \text{Perimeter} = (2x + 3y) + (x + 5y) + (7x - 2y) \)
Group the like terms:
\( = (2x + x + 7x) + (3y + 5y - 2y) \)
\( = 10x + 6y \)
In simple words: To find the perimeter of a triangle, add the expressions for all three sides together by grouping the like terms.

Exam Tip: Remember that perimeter is the total length around a shape. Write the general formula first before substituting the expressions.

Question 8. The two adjacent sides of a rectangle are \( 6a + 9b \) and \( 8a - 4b \). Find its perimeter.
Answer: The length of the rectangle is \( 6a + 9b \) and its breadth is \( 8a - 4b \).
Using the perimeter formula:
\( \text{Perimeter} = 2 \times (\text{Length} + \text{Breadth}) \)
\( = 2 \times [(6a + 9b) + (8a - 4b)] \)
Combine like terms inside the bracket:
\( = 2 \times [(6a + 8a) + (9b - 4b)] \)
\( = 2 \times (14a + 5b) \)
\( = 28a + 10b \)
In simple words: Add the length and the width of the rectangle together, and then double the result to find the total perimeter.

Exam Tip: Always use brackets when substituting expressions into the formula \( 2(l + b) \). This ensures the factor of 2 is correctly distributed to both terms.

Question 9. Subtract the second expression from the first:
(i) \( 2a + b \), \( a + b \)
(ii) \( -2b + 2c \), \( b + 3c \)
(iii) \( 5a + b \), \( -6b + 2a \)
(iv) \( a^3 - 1 + a \), \( 3a - 2a^2 \)
(v) \( p + 2 \), \( 1 \)
(vi) \( x + 2y + z \), \( -x - y - 3z \)
(vii) \( 3a^2 - 8ab - 2b^2 \), \( 3a^2 - 4ab + 6b^2 \)
(viii) \( 4pq - 6p^2 - 2q^2 \), \( 9p^2 \)
(ix) \( 10abc \), \( 2a^2 + 2abc - 4b^2 \)
(x) \( a^2 + ab + c^2 \), \( a^2 - d^2 \)
Answer:
(i) Subtracting the second expression from the first:
\( (2a + b) - (a + b) = 2a + b - a - b = a \)
(ii) \( (-2b + 2c) - (b + 3c) = -2b + 2c - b - 3c = -3b - c \)
(iii) \( (5a + b) - (-6b + 2a) = 5a + b + 6b - 2a = 3a + 7b \)
(iv) \( (a^3 - 1 + a) - (3a - 2a^2) = a^3 - 1 + a - 3a + 2a^2 = a^3 + 2a^2 - 2a - 1 \)
(v) \( (p + 2) - 1 = p + 1 \)
(vi) \( (x + 2y + z) - (-x - y - 3z) = x + 2y + z + x + y + 3z = 2x + 3y + 4z \)
(vii) \( (3a^2 - 8ab - 2b^2) - (3a^2 - 4ab + 6b^2) = 3a^2 - 8ab - 2b^2 - 3a^2 + 4ab - 6b^2 = -4ab - 8b^2 \)
(viii) \( (4pq - 6p^2 - 2q^2) - (9p^2) = 4pq - 6p^2 - 9p^2 - 2q^2 = 4pq - 15p^2 - 2q^2 \)
(ix) \( 10abc - (2a^2 + 2abc - 4b^2) = 10abc - 2abc - 2a^2 + 4b^2 = 8abc - 2a^2 + 4b^2 \)
(x) \( (a^2 + ab + c^2) - (a^2 - d^2) = a^2 + ab + c^2 - a^2 + d^2 = ab + c^2 + d^2 \)
In simple words: When you subtract an expression, change the sign of every single term inside it (plus becomes minus, and minus becomes plus), then combine the like terms.

Exam Tip: The most common mistake in subtraction is forgetting to change the sign of the second term inside the bracket. Always rewrite the expression with the brackets first.

Question 10. Subtract:
(i) \( 4x \) from \( 8 - x \)
(ii) \( -8c \) from \( c + 3d \)
(iii) \( -5a - 2b \) from \( b + 6c \)
(iv) \( 4p + p^2 \) from \( 3p^2 - 8p \)
(v) \( 5a - 3b + 2c \) from \( 4a - b - 2c \)
(vi) \( -xy + yz - zx \) from \( xy - yz + xz \)
(vii) \( 2x^2 - 7xy - y^2 \) from \( 3x^2 - 5xy + 3y^2 \)
(viii) \( a^2 - 3ab - 6b^2 \) from \( 2b^2 - a^2 + 2ab \)
(ix) \( 4x^2 - 5x^2y + y^2 \) from \( -3y^2 + 5xy^2 - 7x^2 - 9x^2y \)
(x) \( 6m^3 + 4m^2 + 7m - 3 \) from \( 3m^3 + 4 \)
Answer:
(i) Subtracting \( 4x \) from \( 8 - x \):
\( (8 - x) - 4x = 8 - x - 4x = 8 - 5x \)
(ii) Subtracting \( -8c \) from \( c + 3d \):
\( (c + 3d) - (-8c) = c + 3d + 8c = 9c + 3d \)
(iii) Subtracting \( -5a - 2b \) from \( b + 6c \):
\( (b + 6c) - (-5a - 2b) = b + 6c + 5a + 2b = 5a + 3b + 6c \)
(iv) Subtracting \( 4p + p^2 \) from \( 3p^2 - 8p \):
\( (3p^2 - 8p) - (4p + p^2) = 3p^2 - 8p - 4p - p^2 = 2p^2 - 12p \)
(v) Subtracting \( 5a - 3b + 2c \) from \( 4a - b - 2c \):
\( (4a - b - 2c) - (5a - 3b + 2c) = 4a - b - 2c - 5a + 3b - 2c = -a + 2b - 4c \)
(vi) Subtracting \( -xy + yz - zx \) from \( xy - yz + xz \):
\( (xy - yz + xz) - (-xy + yz - zx) = xy - yz + xz + xy - yz + zx = 2xy - 2yz + 2xz = 2(xy - yz + xz) \)
(vii) Subtracting \( 2x^2 - 7xy - y^2 \) from \( 3x^2 - 5xy + 3y^2 \):
\( (3x^2 - 5xy + 3y^2) - (2x^2 - 7xy - y^2) = 3x^2 - 5xy + 3y^2 - 2x^2 + 7xy + y^2 = x^2 + 2xy + 4y^2 \)
(viii) Subtracting \( a^2 - 3ab - 6b^2 \) from \( 2b^2 - a^2 + 2ab \):
\( (2b^2 - a^2 + 2ab) - (a^2 - 3ab - 6b^2) = 2b^2 - a^2 + 2ab - a^2 + 3ab + 6b^2 = -2a^2 + 8b^2 + 5ab = 8b^2 + 5ab - 2a^2 \)
(ix) Subtracting \( 4x^2 - 5x^2y + y^2 \) from \( -3y^2 + 5xy^2 - 7x^2 - 9x^2y \):
\( (-3y^2 + 5xy^2 - 7x^2 - 9x^2y) - (4x^2 - 5x^2y + y^2) = -3y^2 + 5xy^2 - 7x^2 - 9x^2y - 4x^2 + 5x^2y - y^2 = -4y^2 + 5xy^2 - 11x^2 - 4x^2y \)
(x) Subtracting \( 6m^3 + 4m^2 + 7m - 3 \) from \( 3m^3 + 4 \):
\( (3m^3 + 4) - (6m^3 + 4m^2 + 7m - 3) = 3m^3 + 4 - 6m^3 - 4m^2 - 7m + 3 = -3m^3 - 4m^2 - 7m + 7 \)
In simple words: When a question asks you to subtract A from B, always write B first, put a minus sign, and then put A inside brackets. This keeps your signs correct when you expand it.

Exam Tip: 'Subtract A from B' means \( B - A \), not \( A - B \). Make sure you write the second expression first in your mathematical statement.

Question 11. Subtract \( -5a^2 - 3a + 1 \) from the sum of \( 4a^2 + 3 - 8a \) and \( 9a - 7 \).
Answer:
First, find the sum of \( 4a^2 + 3 - 8a \) and \( 9a - 7 \):
\( (4a^2 + 3 - 8a) + (9a - 7) = 4a^2 + 3 - 8a + 9a - 7 \)
\( = 4a^2 + a - 4 \)

Next, subtract \( -5a^2 - 3a + 1 \) from the sum obtained above:
\( (4a^2 + a - 4) - (-5a^2 - 3a + 1) \)
\( = 4a^2 + a - 4 + 5a^2 + 3a - 1 \)
Grouping the similar terms together:
\( = (4a^2 + 5a^2) + (a + 3a) - (4 + 1) \)
\( = 9a^2 + 4a - 5 \)
In simple words: First, add the two expressions together. Then, subtract the other expression from your result by changing its signs and combining similar terms.

Exam Tip: Be careful with signs when subtracting. Remember that subtracting a negative term like \( -5a^2 \) turns it into a positive term \( +5a^2 \).

Question 12. By how much does \( 8x^3 - 6x^2 + 9x - 10 \) exceed \( 4x^3 + 2x^2 + 7x - 3 \)?
Answer:
To find how much the first expression exceeds the second, we subtract the second expression from the first:
\( (8x^3 - 6x^2 + 9x - 10) - (4x^3 + 2x^2 + 7x - 3) \)
\( = 8x^3 - 6x^2 + 9x - 10 - 4x^3 - 2x^2 - 7x + 3 \)
Now, group the similar terms:
\( = (8x^3 - 4x^3) + (-6x^2 - 2x^2) + (9x - 7x) + (-10 + 3) \)
\( = 4x^3 - 8x^2 + 2x - 7 \)
In simple words: To see how much larger the first expression is, subtract the second expression from it and combine the matching parts.

Exam Tip: When subtraction is required, wrap the second polynomial in brackets so you do not forget to reverse the sign of each term inside.

Question 13. What must be added to \( 2a^3 + 5a - a^2 - 6 \) to get \( a^2 - a - a^3 + 1 \)?
Answer:
We can find the required expression by subtracting the starting polynomial from the target polynomial:
\( (a^2 - a - a^3 + 1) - (2a^3 + 5a - a^2 - 6) \)
Rearranging the terms in descending powers of \( a \):
\( = (-a^3 + a^2 - a + 1) - (2a^3 - a^2 + 5a - 6) \)
\( = -a^3 + a^2 - a + 1 - 2a^3 + a^2 - 5a + 6 \)
Gathering like terms together:
\( = (-a^3 - 2a^3) + (a^2 + a^2) + (-a - 5a) + (1 + 6) \)
\( = -3a^3 + 2a^2 - 6a + 7 \)
In simple words: To find what to add, subtract the first expression from the second expression.

Exam Tip: Arranging algebraic expressions in descending order of their exponents helps in systematically organizing and simplifying terms without mistakes.

Question 14. What must be subtracted from \( a^2 + b^2 + 2ab \) to get \( -4ab + 2b^2 \)?
Answer:
To obtain the necessary expression, we subtract the target polynomial from the original polynomial:
\( (a^2 + b^2 + 2ab) - (-4ab + 2b^2) \)
\( = a^2 + b^2 + 2ab + 4ab - 2b^2 \)
Reordering and grouping the like terms:
\( = a^2 + (b^2 - 2b^2) + (2ab + 4ab) \)
\( = a^2 - b^2 + 6ab \)
In simple words: To find what to subtract, subtract the target expression from the starting expression.

Exam Tip: Be careful when subtracting terms with exponents. Only combine terms with the exact same variables and exponents, like \( b^2 \) and \( -2b^2 \).

Question 15. Find the excess of \( 4m^2 + 4n^2 + 4p^2 \) over \( m^2 + 3n^2 - 5p^2 \).
Answer:
The excess is found by subtracting the second polynomial from the first:
\( (4m^2 + 4n^2 + 4p^2) - (m^2 + 3n^2 - 5p^2) \)
\( = 4m^2 + 4n^2 + 4p^2 - m^2 - 3n^2 + 5p^2 \)
Grouping similar terms together:
\( = (4m^2 - m^2) + (4n^2 - 3n^2) + (4p^2 + 5p^2) \)
\( = 3m^2 + n^2 + 9p^2 \)
In simple words: Subtract the second group of terms from the first group of terms to see how much larger the first one is.

Exam Tip: Remember that subtracting a negative term like \( -5p^2 \) results in addition, so \( 4p^2 - (-5p^2) = 4p^2 + 5p^2 = 9p^2 \).

Question 16. By how much is \( 3x^3 - 2x^2y + xy^2 - y^3 \) less than \( 4x^3 - 3x^2y - 7xy^2 + 2y^3 \)?
Answer:
To determine how much smaller the first expression is compared to the second, subtract the first from the second:
\( (4x^3 - 3x^2y - 7xy^2 + 2y^3) - (3x^3 - 2x^2y + xy^2 - y^3) \)
\( = 4x^3 - 3x^2y - 7xy^2 + 2y^3 - 3x^3 + 2x^2y - xy^2 + y^3 \)
Gathering the like terms together:
\( = (4x^3 - 3x^3) + (-3x^2y + 2x^2y) + (-7xy^2 - xy^2) + (2y^3 + y^3) \)
\( = x^3 - x^2y - 8xy^2 + 3y^3 \)
In simple words: Subtract the smaller algebraic expression from the larger algebraic expression to find the difference.

Exam Tip: Be extra careful when combining terms with negative coefficients, such as \( -7xy^2 - xy^2 = -8xy^2 \).

Question 17. Subtract the sum of \( 3a^2 - 2a + 5 \) and \( a^2 - 5a - 7 \) from the sum of \( 5a^2 - 9a + 3 \) and \( 2a - a^2 - 1 \).
Answer:
First, calculate the sum of the first two expressions:
\( (3a^2 - 2a + 5) + (a^2 - 5a - 7) \)
\( = 3a^2 + a^2 - 2a - 5a + 5 - 7 \)
\( = 4a^2 - 7a - 2 \)

Next, calculate the sum of the other two expressions:
\( (5a^2 - 9a + 3) + (2a - a^2 - 1) \)
\( = 5a^2 - a^2 - 9a + 2a + 3 - 1 \)
\( = 4a^2 - 7a + 2 \)

Now, subtract the first sum from the second sum:
\( (4a^2 - 7a + 2) - (4a^2 - 7a - 2) \)
\( = 4a^2 - 7a + 2 - 4a^2 + 7a + 2 \)
\( = (4a^2 - 4a^2) + (-7a + 7a) + (2 + 2) \)
\( = 0 + 0 + 4 = 4 \)
In simple words: Find the sum of the first two expressions, then find the sum of the next two. Finally, subtract the first sum from the second sum.

Exam Tip: Notice how almost all terms cancel out during subtraction. Clearly showing each step of grouping like terms helps verify that only the constant \( 4 \) remains.

Question 18. The perimeter of a rectangle is \( 28x^3 + 16x^2 + 8x + 4 \). One of its sides is \( 8x^2 + 4x \). Find the other side
Answer:
The formula for the perimeter of a rectangle is:
\( \text{Perimeter} = 2(l + b) \)
We are given:
\( \text{Perimeter} = 28x^3 + 16x^2 + 8x + 4 \)

Let one of the sides be \( l = 8x^2 + 4x \). Twice this side is:
\( 2l = 2(8x^2 + 4x) = 16x^2 + 8x \)

Since \( \text{Perimeter} = 2l + 2b \), we can find \( 2b \) by subtracting \( 2l \) from the perimeter:
\( 2b = (28x^3 + 16x^2 + 8x + 4) - (16x^2 + 8x) \)
\( = 28x^3 + 16x^2 + 8x + 4 - 16x^2 - 8x \)
\( = 28x^3 + (16x^2 - 16x^2) + (8x - 8x) + 4 \)
\( = 28x^3 + 4 \)

To find the other side \( b \), divide this result by \( 2 \):
\( b = \frac{28x^3 + 4}{2} = 14x^3 + 2 \)
In simple words: The perimeter is twice the sum of both sides. If we double the known side and subtract it from the perimeter, we get twice the unknown side. Divide that by two to get the other side.

Exam Tip: Remember that perimeter is \( 2 \times (\text{length} + \text{width}) \). A common error is subtracting only one side from the perimeter instead of doubling it first or halving the perimeter first.

Question 19. The perimeter of a triangle is \( 14a^2 + 20a + 13 \). Two of its sides are \( 3a^2 + 5a + 1 \) and \( a^2 + 10a - 6 \). Find its third side.
Answer:
The sum of all three sides equals the perimeter of a triangle.
First, let's calculate the sum of the two known sides:
\( (3a^2 + 5a + 1) + (a^2 + 10a - 6) \)
\( = (3a^2 + a^2) + (5a + 10a) + (1 - 6) \)
\( = 4a^2 + 15a - 5 \)

Next, subtract this combined value from the total perimeter to find the third side:
\( \text{Third side} = (14a^2 + 20a + 13) - (4a^2 + 15a - 5) \)
\( = 14a^2 + 20a + 13 - 4a^2 - 15a + 5 \)
Grouping like terms together:
\( = (14a^2 - 4a^2) + (20a - 15a) + (13 + 5) \)
\( = 10a^2 + 5a + 18 \)
In simple words: Add the two known sides of the triangle together, and then subtract that sum from the total perimeter to get the length of the third side.

Exam Tip: Be sure to write brackets around the sum of the two sides when subtracting it from the perimeter, as this changes \( -5 \) to \( +5 \).

Question 20. If \( x = 4a^2 + b^2 - 6ab \), \( y = 3b^2 - 2a^2 + 8ab \) and \( z = 6a^2 + 8b^2 - 6ab \), find:
(i) \( x + y + z \)
(ii) \( x - y - z \)
Answer:
Let's substitute the expressions of \( x \), \( y \), and \( z \):

(i) To find \( x + y + z \):
\( (4a^2 + b^2 - 6ab) + (3b^2 - 2a^2 + 8ab) + (6a^2 + 8b^2 - 6ab) \)
Gathering identical terms together:
\( = (4a^2 - 2a^2 + 6a^2) + (b^2 + 3b^2 + 8b^2) + (-6ab + 8ab - 6ab) \)
\( = 8a^2 + 12b^2 - 4ab \)

(ii) To find \( x - y - z \):
\( (4a^2 + b^2 - 6ab) - (3b^2 - 2a^2 + 8ab) - (6a^2 + 8b^2 - 6ab) \)
Open the brackets, making sure to reverse the signs of terms inside subtracted brackets:
\( = 4a^2 + b^2 - 6ab - 3b^2 + 2a^2 - 8ab - 6a^2 - 8b^2 + 6ab \)
Combining similar terms:
\( = (4a^2 + 2a^2 - 6a^2) + (b^2 - 3b^2 - 8b^2) + (-6ab - 8ab + 6ab) \)
\( = -10b^2 - 8ab \)
In simple words: Replace the variables \( x \), \( y \), and \( z \) with their algebraic values. For addition, group and combine identical terms. For subtraction, flip the signs of all terms inside the subtracted groups first.

Exam Tip: Be vigilant with negative signs when working out \( x - y - z \). A very common error is forgetting to change the sign of the trailing terms inside the parentheses.

Question 21. If \( m = 9x^2 - 4xy + 5y^2 \) and \( n = -3x^2 + 2xy - y^2 \), find:
(i) \( 2m - n \)
(ii) \( m + 2n \)
(iii) \( m - 3n \)
Answer:
Substitute the values of \( m \) and \( n \) in each case:

(i) Calculating \( 2m - n \):
\( 2(9x^2 - 4xy + 5y^2) - (-3x^2 + 2xy - y^2) \)
\( = 18x^2 - 8xy + 10y^2 + 3x^2 - 2xy + y^2 \)
Combining like terms:
\( = (18x^2 + 3x^2) + (-8xy - 2xy) + (10y^2 + y^2) \)
\( = 21x^2 - 10xy + 11y^2 \)

(ii) Calculating \( m + 2n \):
\( (9x^2 - 4xy + 5y^2) + 2(-3x^2 + 2xy - y^2) \)
\( = 9x^2 - 4xy + 5y^2 - 6x^2 + 4xy - 2y^2 \)
Combining like terms:
\( = (9x^2 - 6x^2) + (-4xy + 4xy) + (5y^2 - 2y^2) \)
\( = 3x^2 + 3y^2 \)

(iii) Calculating \( m - 3n \):
\( (9x^2 - 4xy + 5y^2) - 3(-3x^2 + 2xy - y^2) \)
\( = 9x^2 - 4xy + 5y^2 + 9x^2 - 6xy + 3y^2 \)
Combining like terms:
\( = (9x^2 + 9x^2) + (-4xy - 6xy) + (5y^2 + 3y^2) \)
\( = 18x^2 - 10xy + 8y^2 \)
In simple words: For each expression, multiply the groups of terms by their given coefficients, then combine the similar \( x^2 \), \( xy \), and \( y^2 \) terms.

Exam Tip: In part (ii), notice that the \( xy \) terms cancel out entirely since \( -4xy + 4xy = 0 \). Make sure to show this simplification clearly in your working.

Question 22. Simplify:
(i) \( 3x + 5(2x + 6) - 7x \)
(ii) \( 3(4y - 10) + 2(y - 1) \)
(iii) \( -(7 + 6x) - 7(x + 2) \)
(iv) \( x - (x - y) - y - (y - x) \)
(v) \( 4x + 7y - [5y - 8] - 2x \)
(vi) \( -2m + 5 + 4(m - 3) \)
(vii) \( 2x - y + 5 - (x - y) \)
(viii) \( 2(x - y) - (x - 8) \)
(ix) \( 4(3x - 8) - 3(5x + 3) - 2(6x - 8) \)
(x) \( 5(x - 4) - 3(x - 4) + 7(x - 4) \)
Answer:
Expanding brackets and grouping like terms for each expression:

(i) \( 3x + 5(2x + 6) - 7x \)
\( = 3x + 10x + 30 - 7x \)
\( = 6x + 30 \)

(ii) \( 3(4y - 10) + 2(y - 1) \)
\( = 12y - 30 + 2y - 2 \)
\( = 14y - 32 \)

(iii) \( -(7 + 6x) - 7(x + 2) \)
\( = -7 - 6x - 7x - 14 \)
\( = -13x - 21 \)

(iv) \( x - (x - y) - y - (y - x) \)
\( = x - x + y - y - y + x \)
\( = x - y \)

(v) \( 4x + 7y - [5y - 8] - 2x \)
\( = 4x + 7y - 5y + 8 - 2x \)
\( = 2x + 2y + 8 \)

(vi) \( -2m + 5 + 4(m - 3) \)
\( = -2m + 5 + 4m - 12 \)
\( = 2m - 7 \)

(vii) \( 2x - y + 5 - (x - y) \)
\( = 2x - y + 5 - x + y \)
\( = x + 5 \)

(viii) \( 2(x - y) - (x - 8) \)
\( = 2x - 2y - x + 8 \)
\( = x - 2y + 8 \)

(ix) \( 4(3x - 8) - 3(5x + 3) - 2(6x - 8) \)
\( = 12x - 32 - 15x - 9 - 12x + 16 \)
\( = -15x - 25 \)

(x) \( 5(x - 4) - 3(x - 4) + 7(x - 4) \)
\( = (5 - 3 + 7)(x - 4) \)
\( = 9(x - 4) \)
\( = 9x - 36 \)
In simple words: Multiply the factor outside each parenthesis with every term inside. Then, group the identical variables and numbers together to combine them.

Exam Tip: Be very careful when expanding negative signs outside brackets. A negative sign reverses every sign inside the bracket, such as \( -[5y - 8] = -5y + 8 \).

Exercise 11(C)

Question 1. Multiply:
(i) \( 3x \), \( 5x^2y \) and \( 2y \)
(ii) \( 5 \), \( 3a \) and \( 2ab^2 \)
(iii) \( 5x + 2y \) and \( 3xy \)
(iv) \( 6a - 5b \) and \( -2a \)
(v) \( 4a + 5b \) and \( 4a - 5b \)
(vi) \( 9xy + 2y^2 \) and \( 2x - 3y \)
(vii) \( -3m^2n + 5mn - 4mn^2 \) and \( 6m^2n \)
(viii) \( 6xy^2 - 7x^2y^2 + 10x^3 \) and \( -3x^2y^3 \)
Answer:
Performing the multiplication for each sub-part:

(i) Multiplying the monomials:
\( 3x \times 5x^2y \times 2y = (3 \times 5 \times 2) \times (x \times x^2) \times (y \times y) \)
\( = 30x^3y^2 \)

(ii) Multiplying the monomials:
\( 5 \times 3a \times 2ab^2 = (5 \times 3 \times 2) \times (a \times a) \times b^2 \)
\( = 30a^2b^2 \)

(iii) Expanding by distributing \( 3xy \):
\( (5x + 2y) \times 3xy = 5x(3xy) + 2y(3xy) \)
\( = 15x^2y + 6xy^2 \)

(iv) Distributing \( -2a \):
\( (6a - 5b) \times (-2a) = 6a(-2a) - 5b(-2a) \)
\( = -12a^2 + 10ab \)

(v) Expanding the product of two binomials:
\( (4a + 5b)(4a - 5b) = 4a(4a - 5b) + 5b(4a - 5b) \)
\( = 16a^2 - 20ab + 20ab - 25b^2 \)
\( = 16a^2 - 25b^2 \)

(vi) Expanding the product of binomials:
\( (9xy + 2y^2)(2x - 3y) = 9xy(2x - 3y) + 2y^2(2x - 3y) \)
\( = 18x^2y - 27xy^2 + 4xy^2 - 6y^3 \)
\( = 18x^2y - 23xy^2 - 6y^3 \)

(vii) Distributing \( 6m^2n \) over each term of the trinomial:
\( 6m^2n(-3m^2n + 5mn - 4mn^2) \)
\( = 6m^2n \times (-3m^2n) + 6m^2n \times (5mn) + 6m^2n \times (-4mn^2) \)
\( = -18m^4n^2 + 30m^3n^2 - 24m^3n^3 \)

(viii) Distributing \( -3x^2y^3 \) over each term:
\( -3x^2y^3(6xy^2 - 7x^2y^2 + 10x^3) \)
\( = -3x^2y^3 \times (6xy^2) - 3x^2y^3 \times (-7x^2y^2) - 3x^2y^3 \times (10x^3) \)
\( = -18x^3y^5 + 21x^4y^5 - 30x^5y^3 \)
In simple words: When multiplying terms, multiply the numbers together first. Then, add the exponents of identical variables. For brackets, multiply the term outside with every single term inside.

Exam Tip: Remember the laws of exponents: when multiplying the same bases, add their powers (e.g., \( x^2 \times x = x^3 \)). Keep track of the signs carefully.

Question 2. Copy and complete the following multiplications:
(i) \( (3a + 2b) \times (-3xy) \)
(ii) \( (9x - 5y) \times (-3xy) \)
(iii) \( (3xy - 2x^2 - 6x) \times (-5x^2y) \)
(iv) \( (a + b) \times (a + b) \)
(v) \( (ax - b) \times (2ax + 2b^2) \)
(vi) \( (2a - b + 3c) \times (2a - 4b) \)
(vii) \( (3m^2 + 6m - 2n) \times (5n - 3m) \)
(viii) \( (6 - 3x + 2x^2) \times (1 + 5x - x^2) \)
(ix) \( (4x^3 - 10x^2 + 6x - 8) \times (3 + 2x - x^2) \)
Answer:
Completing the vertical multiplications by multiplying each term of the bottom expression with the top expression:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(ix)

In simple words: Write the multiplication step-by-step. Multiply each term of the second expression by every term of the first expression, align the like terms under each other, and then add them up.
Exam Tip: Aligning like terms in columns makes vertical multiplication much easier and helps prevent mistakes when adding terms together.

Question 3. Evaluate:
(i) \( (c + 5)(c - 3) \)
(ii) \( (3c - 5d)(4c - 6d) \)
(iii) \( \left(\frac{1}{2}a + \frac{1}{2}b\right)\left(\frac{1}{2}a - \frac{1}{2}b\right) \)
(iv) \( (a^2 + 2ab + b^2)(a + b) \)
(v) \( (3x - 1)(4x^3 - 2x^2 + 6x - 3) \)
(vi) \( (4m - 2)(m^2 + 5m - 6) \)
(vii) \( (8 - 12x + 7x^2 - 6x^3)(5 - 2x) \)
(viii) \( (4x^2 - 4x + 1)(2x^3 - 3x^2 + 2) \)
(ix) \( (6p^2 - 8pq + 2q^2)(-5p) \)
(x) \( -4y(15x + 12y - 8z)(x - 2y) \)
(xi) \( (a^2 + b^2 + c^2 - ab - bc - ca)(a + b + c) \)
Answer:
Expanding and simplifying each product:

(i) \( (c + 5)(c - 3) = c(c - 3) + 5(c - 3) \)
\( = c^2 - 3c + 5c - 15 \)
\( = c^2 + 2c - 15 \)

(ii) \( (3c - 5d)(4c - 6d) = 3c(4c - 6d) - 5d(4c - 6d) \)
\( = 12c^2 - 18cd - 20cd + 30d^2 \)
\( = 12c^2 - 38cd + 30d^2 \)

(iii) \( \left(\frac{1}{2}a + \frac{1}{2}b\right)\left(\frac{1}{2}a - \frac{1}{2}b\right) = \frac{1}{2}a\left(\frac{1}{2}a - \frac{1}{2}b\right) + \frac{1}{2}b\left(\frac{1}{2}a - \frac{1}{2}b\right) \)
\( = \frac{1}{4}a^2 - \frac{1}{4}ab + \frac{1}{4}ab - \frac{1}{4}b^2 \)
\( = \frac{1}{4}a^2 - \frac{1}{4}b^2 \)

(iv) \( (a^2 + 2ab + b^2)(a + b) = a(a^2 + 2ab + b^2) + b(a^2 + 2ab + b^2) \)
\( = a^3 + 2a^2b + ab^2 + a^2b + 2ab^2 + b^3 \)
\( = a^3 + 3a^2b + 3ab^2 + b^3 \)

(v) \( (3x - 1)(4x^3 - 2x^2 + 6x - 3) = 3x(4x^3 - 2x^2 + 6x - 3) - 1(4x^3 - 2x^2 + 6x - 3) \)
\( = 12x^4 - 6x^3 + 18x^2 - 9x - 4x^3 + 2x^2 - 6x + 3 \)
\( = 12x^4 - 10x^3 + 20x^2 - 15x + 3 \)

(vi) \( (4m - 2)(m^2 + 5m - 6) = 4m(m^2 + 5m - 6) - 2(m^2 + 5m - 6) \)
\( = 4m^3 + 20m^2 - 24m - 2m^2 - 10m + 12 \)
\( = 4m^3 + 18m^2 - 34m + 12 \)

(vii) \( (8 - 12x + 7x^2 - 6x^3)(5 - 2x) = 5(8 - 12x + 7x^2 - 6x^3) - 2x(8 - 12x + 7x^2 - 6x^3) \)
\( = 40 - 60x + 35x^2 - 30x^3 - 16x + 24x^2 - 14x^3 + 12x^4 \)
\( = 40 - 76x + 59x^2 - 44x^3 + 12x^4 \)

(viii) \( (4x^2 - 4x + 1)(2x^3 - 3x^2 + 2) = 4x^2(2x^3 - 3x^2 + 2) - 4x(2x^3 - 3x^2 + 2) + 1(2x^3 - 3x^2 + 2) \)
\( = 8x^5 - 12x^4 + 8x^2 - 8x^4 + 12x^3 - 8x + 2x^3 - 3x^2 + 2 \)
\( = 8x^5 - 20x^4 + 14x^3 + 5x^2 - 8x + 2 \)

(ix) \( (6p^2 - 8pq + 2q^2)(-5p) = -5p(6p^2 - 8pq + 2q^2) \)
\( = -30p^3 + 40p^2q - 10pq^2 \)

(x) \( -4y(15x + 12y - 8z)(x - 2y) \)
First, multiply \( -4y(x - 2y) = -4xy + 8y^2 \):
\( = (-4xy + 8y^2)(15x + 12y - 8z) \)
\( = -4xy(15x + 12y - 8z) + 8y^2(15x + 12y - 8z) \)
\( = -60x^2y - 48xy^2 + 32xyz + 120xy^2 + 96y^3 - 64y^2z \)
\( = -60x^2y + 72xy^2 - 64y^2z + 96y^3 + 32xyz \)

(xi) \( (a^2 + b^2 + c^2 - ab - bc - ca)(a + b + c) \)
\( = a(a^2 + b^2 + c^2 - ab - bc - ca) + b(a^2 + b^2 + c^2 - ab - bc - ca) + c(a^2 + b^2 + c^2 - ab - bc - ca) \)
\( = a^3 + ab^2 + ac^2 - a^2b - abc - ca^2 + a^2b + b^3 + bc^2 - ab^2 - b^2c - abc + ca^2 + b^2c + c^3 - abc - bc^2 - c^2a \)
Simplifying and cancelling identical positive and negative terms:
\( = a^3 + b^3 + c^3 - 3abc \)
In simple words: Multiply each term of the first expression by every term of the second expression. After expanding, group the terms that are alike and combine them to get the final simplified expression.

Exam Tip: For longer expansions like part (xi), cross out terms that cancel each other (e.g., \( +ab^2 \) and \( -ab^2 \)) to ensure you do not miss any remaining terms.

Question 4. Evaluate:
(i) \( (a + b)(a - b) \)
(ii) \( (a^2 + b^2)(a + b)(a - b) \); using the result of (i).
(iii) \( (a^4 + b^4)(a^2 + b^2)(a + b)(a - b) \); using the result of (ii).
Answer:
Evaluating each sub-part step-by-step:

(i) \( (a + b)(a - b) = a(a - b) + b(a - b) \)
\( = a^2 - ab + ab - b^2 \)
\( = a^2 - b^2 \)

(ii) Using the result from (i), we can substitute \( a^2 - b^2 \) in place of \( (a + b)(a - b) \):
\( (a^2 + b^2)[(a + b)(a - b)] = (a^2 + b^2)(a^2 - b^2) \)
\( = a^2(a^2 - b^2) + b^2(a^2 - b^2) \)
\( = a^4 - a^2b^2 + a^2b^2 - b^4 \)
\( = a^4 - b^4 \)

(iii) Using the result from (ii), we can substitute \( a^4 - b^4 \) in place of \( (a^2 + b^2)(a + b)(a - b) \):
\( (a^4 + b^4)[(a^2 + b^2)(a + b)(a - b)] = (a^4 + b^4)(a^4 - b^4) \)
\( = a^4(a^4 - b^4) + b^4(a^4 - b^4) \)
\( = a^8 - a^4b^4 + a^4b^4 - b^8 \)
\( = a^8 - b^8 \)
In simple words: Multiplying a sum by a difference of the same two terms always results in the difference of their squares. We can use this shortcut repeatedly to solve larger chains of multiplication.

Exam Tip: This illustrates the classic algebraic identity \( (x + y)(x - y) = x^2 - y^2 \). Applying this identity directly can save a lot of time and avoid expansion errors.

Question 5. Evaluate:
(i) \( (3x - 2y)(4x + 3y) \)
(ii) \( (3x - 2y)(4x + 3y)(8x - 5y) \)
(iii) \( (a + 5)(3a - 2)(5a + 1) \)
(iv) \( (a + 1)(a^2 - a + 1) \) and \( (a - 1)(a^2 + a + 1) \); and then: \( (a + 1)(a^2 - a + 1) + (a - 1)(a^2 + a + 1) \)
(v) \( (5m - 2n)(5m + 2n)(25m^2 + 4n^2) \)
Answer:
Solving each sub-part sequentially:

(i) \( (3x - 2y)(4x + 3y) = 3x(4x + 3y) - 2y(4x + 3y) \)
\( = 12x^2 + 9xy - 8xy - 6y^2 \)
\( = 12x^2 + xy - 6y^2 \)

(ii) Substitute the result from part (i):
\( (12x^2 + xy - 6y^2)(8x - 5y) \)
\( = 8x(12x^2 + xy - 6y^2) - 5y(12x^2 + xy - 6y^2) \)
\( = 96x^3 + 8x^2y - 48xy^2 - 60x^2y - 5xy^2 + 30y^3 \)
Combining like terms:
\( = 96x^3 - 52x^2y - 53xy^2 + 30y^3 \)

(iii) First, let's multiply the first two brackets:
\( (a + 5)(3a - 2) = a(3a - 2) + 5(3a - 2) \)
\( = 3a^2 - 2a + 15a - 10 = 3a^2 + 13a - 10 \)
Now multiply this result with the third bracket:
\( (3a^2 + 13a - 10)(5a + 1) \)
\( = 5a(3a^2 + 13a - 10) + 1(3a^2 + 13a - 10) \)
\( = 15a^3 + 65a^2 - 50a + 3a^2 + 13a - 10 \)
Grouping like terms:
\( = 15a^3 + 68a^2 - 37a - 10 \)

(iv) Evaluate the two separate products first:
\( (a + 1)(a^2 - a + 1) = a(a^2 - a + 1) + 1(a^2 - a + 1) \)
\( = a^3 - a^2 + a + a^2 - a + 1 = a^3 + 1 \)
And for the second product:
\( (a - 1)(a^2 + a + 1) = a(a^2 + a + 1) - 1(a^2 + a + 1) \)
\( = a^3 + a^2 + a - a^2 - a - 1 = a^3 - 1 \)
Now add both results together:
\( (a^3 + 1) + (a^3 - 1) = 2a^3 \)

(v) Use the identity \( (x - y)(x + y) = x^2 - y^2 \) on the first two terms:
\( (5m - 2n)(5m + 2n) = (5m)^2 - (2n)^2 = 25m^2 - 4n^2 \)
Now multiply this with the final bracket:
\( (25m^2 - 4n^2)(25m^2 + 4n^2) \)
Applying the same difference of squares identity again:
\( = (25m^2)^2 - (4n^2)^2 \)
\( = 625m^4 - 16n^4 \)
In simple words: When multiplying three brackets together, multiply the first two first. Then, take that result and multiply it by the last bracket. Look out for differences of squares shortcuts like in part (v) to make the calculations easier.

Exam Tip: Recognizing standard expansions such as the sum of cubes \( (a+1)(a^2-a+1) = a^3+1 \) and difference of cubes \( (a-1)(a^2+a+1) = a^3-1 \) can simplify the working immensely.

Question 6. Multiply:
(i) \( mn^4 \), \( m^3n \) and \( 5m^2n^3 \)
(ii) \( 2mnpq \), \( 4mnpq \) and \( 5mnpq \)
(iii) \( pq - pm \) and \( p^2m \)
(iv) \( x^3 - 3y^3 \) and \( 4x^2y^2 \)
(v) \( a^3 - 4ab \) and \( 2a^2b \)
(vi) \( x^2 + 5yx - 3y^2 \) and \( 2x^2y \)
Answer:
(i) Multiplication of the given terms:
\( 5m^2n^3 \times mn^4 \times m^3n \)
\( \implies 5m^{(2 + 1 + 3)}n^{(3 + 4 + 1)} \)
\( = 5m^6n^8 \)

(ii) Multiplication of the terms:
\( 5mnpq \times 2mnpq \times 4mnpq \)
\( \implies 5 \times 2 \times 4 \times m^{(1 + 1 + 1)}n^{(1 + 1 + 1)}p^{(1 + 1 + 1)}q^{(1 + 1 + 1)} \)
\( \implies 40m^3n^3p^3q^3 \)

(iii) Multiplication of the terms:
\( p^2m \times (pq - pm) \)
\( \implies p^3qm - p^3m^2 \)

(iv) Multiplying the expression by the term:
\( 4x^2y^2 \times (x^3 - 3y^3) \)
\( \implies 4x^5y^2 - 12x^2y^5 \)

(v) Multiplication of the terms:
\( 2a^2b \times (a^3 - 4ab) \)
\( \implies 2a^5b - 8a^3b^2 \)

(vi) Multiplying the expression by the term:
\( 2x^2y \times (x^2 + 5yx - 3y^2) \)
\( \implies 2x^4y + 10x^3y^2 - 6x^2y^3 \)
In simple words: When you multiply algebraic terms, you multiply the numbers together and add the powers of the matching letters. If there is a bracket, make sure you multiply the outside term with every single term inside the bracket.

Exam Tip: Be careful with signs and remember to add the exponents of identical variables when multiplying. A common mistake is multiplying exponents instead of adding them.

Question 7. Multiply:
(i) \( (2x + 3y)(2x + 3y) \)
(ii) \( (2x - 3y)(2x + 3y) \)
(iii) \( (2x + 3y)(2x - 3y) \)
(iv) \( (2x - 3y)(2x - 3y) \)
(v) \( (-2x + 3y)(2x - 3y) \)
(vi) \( (xy + 2b)(xy - 2b) \)
(vii) \( (x - a)(x + 3b) \)
(viii) \( (2x + 5y + 6)(3x + y - 8) \)
(ix) \( (3x - 5y + 2)(5x - 4y - 3) \)
(x) \( (6x - 2y)(3x - y) \)
(xi) \( (1 + 6x^2 - 4x^3)(-1 + 3x - 3x^2) \)
Answer:
(i) Expanding the brackets:
\( 2x(2x + 3y) + 3y(2x + 3y) \)
\( \implies 4x^2 + 6xy + 6xy + 9y^2 \)
\( \implies 4x^2 + 12xy + 9y^2 \)

(ii) Expanding the terms:
\( 2x(2x + 3y) - 3y(2x + 3y) \)
\( \implies 2x \times 2x + 2x \times 3y - 3y \times 2x - 3y \times 3y \)
\( \implies 4x^2 + 6xy - 6xy - 9y^2 \)
\( \implies 4x^2 + 0 - 9y^2 \)
\( \implies 4x^2 - 9y^2 \)

(iii) Expanding the multiplication:
\( 2x(2x - 3y) + 3y(2x - 3y) \)
\( \implies 2x \times 2x - 2x \times 3y + 3y \times 2x - 3y \times 3y \)
\( \implies 4x^2 - 6xy + 6xy - 9y^2 \)
\( \implies 4x^2 - 0 - 9y^2 \)
\( \implies 4x^2 - 9y^2 \)

(iv) Expanding the squared term:
\( 2x(2x - 3y) - 3y(2x - 3y) \)
\( \implies 2x \times 2x - 2x \times 3y - 3y \times 2x + 3y \times 3y \)
\( \implies 4x^2 - 6xy - 6xy + 9y^2 \)
\( \implies 4x^2 - 12xy + 9y^2 \)

(v) Expanding the negative terms:
\( -2x(2x - 3y) + 3y(2x - 3y) \)
\( \implies -4x^2 + 6xy + 6xy - 9y^2 \)
\( \implies -4x^2 + 12xy - 9y^2 \)

(vi) Expanding with two variables:
\( xy(xy - 2b) + 2b(xy - 2b) \)
\( \implies x^2y^2 - 2bxy + 2bxy - 4b^2 \)
\( \implies x^2y^2 - 4b^2 \)

(vii) Multiplying linear terms:
\( x(x + 3b) - a(x + 3b) \)
\( \implies x^2 + 3bx - ax - 3ab \)

(viii) Multiplying trinomials:
\( 2x(3x + y - 8) + 5y(3x + y - 8) + 6(3x + y - 8) \)
\( \implies 6x^2 + 2xy - 16x + 15xy + 5y^2 - 40y + 18x + 6y - 48 \)
\( \implies 6x^2 + 2xy + 15xy - 16x + 18x + 5y^2 - 40y + 6y - 48 \)
\( \implies 6x^2 + 17xy + 2x + 5y^2 - 34y - 48 \)

(ix) Multiplying trinomial terms:
\( 3x(5x - 4y - 3) - 5y(5x - 4y - 3) + 2(5x - 4y - 3) \)
\( \implies 15x^2 - 12xy - 9x - 25xy + 20y^2 + 15y + 10x - 8y - 6 \)
\( \implies 15x^2 - 12xy - 25xy - 9x + 10x + 20y^2 + 15y - 8y - 6 \)
\( \implies 15x^2 - 37xy + x + 20y^2 + 7y - 6 \)

(x) Multiplying the binomials:
\( 6x(3x - y) - 2y(3x - y) \)
\( \implies 18x^2 - 6xy - 6xy + 2y^2 \)
\( \implies 18x^2 - 12xy + 2y^2 \)

(xi) Multiplying algebraic expressions:
\( 1(-1 + 3x - 3x^2) + 6x^2(-1 + 3x - 3x^2) - 4x^3(-1 + 3x - 3x^2) \)
\( \implies -1 + 3x - 3x^2 - 6x^2 + 18x^3 - 18x^4 + 4x^3 - 12x^4 + 12x^5 \)
\( \implies -1 + 3x - 9x^2 + 22x^3 - 30x^4 + 12x^5 \)
In simple words: To multiply two expressions, take each part of the first expression and multiply it by everything in the second expression. After that, group the terms that are alike to get the final simplified answer.

Exam Tip: Be very careful when distributing a negative sign across a bracket. For example, negative times negative becomes positive.

Exercise 11(D)

Question 1. Divide:
(i) \( -16ab^2c \) by \( 6abc \)
(ii) \( 25x^2y \) by \( -5y^2 \)
(iii) \( 8x + 24 \) by \( 4 \)
(iv) \( 4a^2 - a \) by \( -a \)
(v) \( 8m - 16 \) by \( -8 \)
(vi) \( -50 + 40p \) by \( 10p \)
(vii) \( 4x^3 - 2x^2 \) by \( -x \)
(viii) \( 10a^3 - 15a^2b \) by \( -5a^2 \)
(ix) \( 12x^3y - 8x^2y^2 + 4x^2y^3 \) by \( 4xy \)
(x) \( 9a^4b - 15a^3b^2 + 12a^2b^3 \) by \( -3a^2b \)
Answer:
(i) Dividing the terms:
\( \frac{-16ab^2c}{6abc} = -\frac{8}{3}b \)

(ii) Performing the division:
\( \frac{25x^2y}{-5y^2} = -5\frac{x^2}{y} \)

(iii) Separating each term in the numerator:
\( \frac{8x + 24}{4} = \frac{8x}{4} + \frac{24}{4} = 2x + 6 \)

(iv) Dividing both terms by \( -a \):
\( \frac{4a^2 - a}{-a} = \frac{4a^2}{-a} - \frac{a}{-a} = -4a + 1 \)

(v) Dividing the binomial:
\( \frac{8m - 16}{-8} = \frac{8m}{-8} - \frac{16}{-8} = -m + 2 \)

(vi) Splitting the numerator:
\( \frac{-50 + 40p}{10p} = \frac{-50}{10p} + \frac{40p}{10p} = -\frac{5}{p} + 4 \)

(vii) Dividing by \( -x \):
\( \frac{4x^3 - 2x^2}{-x} = \frac{4x^3}{-x} - \frac{2x^2}{-x} = -4x^2 + 2x \)

(viii) Dividing each part of the expression:
\( \frac{10a^3 - 15a^2b}{-5a^2} = \frac{10a^3}{-5a^2} - \frac{15a^2b}{-5a^2} = -2a + 3b \)

(ix) Dividing the trinomial by the monomial:
\( \frac{12x^3y - 8x^2y^2 + 4x^2y^3}{4xy} = \frac{12x^3y}{4xy} - \frac{8x^2y^2}{4xy} + \frac{4x^2y^3}{4xy} = 3x^2 - 2xy + xy^2 \)

(x) Dividing each term of the polynomial:
\( \frac{9a^4b - 15a^3b^2 + 12a^2b^3}{-3a^2b} = \frac{9a^4b}{-3a^2b} - \frac{15a^3b^2}{-3a^2b} + \frac{12a^2b^3}{-3a^2b} = -3a^2 + 5ab - 4b^2 \)
In simple words: When dividing an expression with multiple terms by a single term, divide each part of the top expression by the bottom term one by one. Remember to subtract the exponents of the same variables.

Exam Tip: Be extra careful with minus signs. When you divide a negative term by another negative term, the result becomes positive.

Question 2. Divide:
(i) \( n^2 - 2n + 1 \) by \( n - 1 \)
(ii) \( m^2 - 2mn + n^2 \) by \( m - n \)
(iii) \( 4a^2 + 4a + 1 \) by \( 2a + 1 \)
(iv) \( p^2 + 4p + 4 \) by \( p + 2 \)
(v) \( x^2 + 4xy + 4y^2 \) by \( x + 2y \)
(vi) \( 2a^2 - 11a + 12 \) by \( a - 4 \)
(vii) \( 6x^2 + 5x - 6 \) by \( 2x + 3 \)
(viii) \( 8a^2 + 4a - 60 \) by \( 2a - 5 \)
(ix) \( 9x^2 - 24xy + 16y^2 \) by \( 3x - 4y \)
(x) \( 15x^2 + 31xy + 14y^2 \) by \( 5x + 7y \)
(xi) \( 35a^3 + 3a^2b - 2ab^2 \) by \( 5a - b \)
(xii) \( 6x^3 + 5x^2 - 21x + 10 \) by \( 3x - 2 \)
Answer:
(i) Division using polynomial long division:
\[ \begin{array}{r} n - 1 \\ n - 1 \overline{\smash{)} \ n^2 - 2n + 1} \\ \underline{n^2 - n\phantom{ + 0}} \\ -n + 1 \\ \underline{-n + 1} \\ 0 \\ \end{array} \]
The quotient is \( n - 1 \).

(ii) Long division:
\[ \begin{array}{r} m - n \\ m - n \overline{\smash{)} \ m^2 - 2mn + n^2} \\ \underline{m^2 - mn\phantom{ + 0}} \\ -mn + n^2 \\ \underline{-mn + n^2} \\ 0 \\ \end{array} \]
The quotient is \( m - n \).

(iii) Long division:
\[ \begin{array}{r} 2a + 1 \\ 2a + 1 \overline{\smash{)} \ 4a^2 + 4a + 1} \\ \underline{4a^2 + 2a\phantom{ + 0}} \\ 2a + 1 \\ \underline{2a + 1} \\ 0 \\ \end{array} \]
The quotient is \( 2a + 1 \).

(iv) Long division:
\[ \begin{array}{r} p + 2 \\ p + 2 \overline{\smash{)} \ p^2 + 4p + 4} \\ \underline{p^2 + 2p\phantom{ + 0}} \\ 2p + 4 \\ \underline{2p + 4} \\ 0 \\ \end{array} \]
The quotient is \( p + 2 \).

(v) Long division:
\[ \begin{array}{r} x + 2y \\ x + 2y \overline{\smash{)} \ x^2 + 4xy + 4y^2} \\ \underline{x^2 + 2xy\phantom{ + 0}} \\ 2xy + 4y^2 \\ \underline{2xy + 4y^2} \\ 0 \\ \end{array} \]
The quotient is \( x + 2y \).

(vi) Long division:
\[ \begin{array}{r} 2a - 3 \\ a - 4 \overline{\smash{)} \ 2a^2 - 11a + 12} \\ \underline{2a^2 - 8a\phantom{ + 0}} \\ -3a + 12 \\ \underline{-3a + 12} \\ 0 \\ \end{array} \]
The quotient is \( 2a - 3 \).

(vii) Long division:
\[ \begin{array}{r} 3x - 2 \\ 2x + 3 \overline{\smash{)} \ 6x^2 + 5x - 6} \\ \underline{6x^2 + 9x\phantom{ + 0}} \\ -4x - 6 \\ \underline{-4x - 6} \\ 0 \\ \end{array} \]
The quotient is \( 3x - 2 \).

(viii) Long division:
\[ \begin{array}{r} 4a + 12 \\ 2a - 5 \overline{\smash{)} \ 8a^2 + 4a - 60} \\ \underline{8a^2 - 20a\phantom{ + 0}} \\ 24a - 60 \\ \underline{24a - 60} \\ 0 \\ \end{array} \]
The quotient is \( 4a + 12 \).

(ix) Long division:
\[ \begin{array}{r} 3x - 4y \\ 3x - 4y \overline{\smash{)} \ 9x^2 - 24xy + 16y^2} \\ \underline{9x^2 - 12xy\phantom{ + 0}} \\ -12xy + 16y^2 \\ \underline{-12xy + 16y^2} \\ 0 \\ \end{array} \]
The quotient is \( 3x - 4y \).

(x) Long division:
\[ \begin{array}{r} 3x + 2y \\ 5x + 7y \overline{\smash{)} \ 15x^2 + 31xy + 14y^2} \\ \underline{15x^2 + 21xy\phantom{ + 0}} \\ 10xy + 14y^2 \\ \underline{10xy + 14y^2} \\ 0 \\ \end{array} \]
The quotient is \( 3x + 2y \).

(xi) Long division:
\[ \begin{array}{r} 7a^2 + 2ab \\ 5a - b \overline{\smash{)} \ 35a^3 + 3a^2b - 2ab^2} \\ \underline{35a^3 - 7a^2b\phantom{ + 0}} \\ 10a^2b - 2ab^2 \\ \underline{10a^2b - 2ab^2} \\ 0 \\ \end{array} \]
The quotient is \( 7a^2 + 2ab \).

(xii) Long division:
\[ \begin{array}{r} 2x^2 + 3x - 5 \\ 3x - 2 \overline{\smash{)} \ 6x^3 + 5x^2 - 21x + 10} \\ \underline{6x^3 - 4x^2\phantom{ + 0 + 0}} \\ 9x^2 - 21x \\ \underline{9x^2 - 6x\phantom{ + 0}} \\ -15x + 10 \\ \underline{-15x + 10} \\ 0 \\ \end{array} \]
The quotient is \( 2x^2 + 3x - 5 \).
In simple words: Polynomial long division works just like long division with numbers. Find what to multiply the first term of the divisor by to get the first term of the dividend, write it on top, multiply, subtract, and bring down the next term.

Exam Tip: Always make sure both the dividend and the divisor are written in decreasing order of their powers before you start dividing.

Question 3. The area of a rectangle is \( 6x^2 - 4xy - 10y^2 \) square units and its length is \( 2x + 2y \) units. Find its breadth.
Answer:
Given:
Area of the rectangle = \( 6x^2 - 4xy - 10y^2 \) square units
Length of the rectangle = \( 2x + 2y \) units

We know that:
\( \text{Breadth} = \frac{\text{Area}}{\text{Length}} \)

\( \text{Breadth} = \frac{6x^2 - 4xy - 10y^2}{2x + 2y} \)

Performing long division to find the breadth:
\[ \begin{array}{r} 3x - 5y \\ 2x + 2y \overline{\smash{)} \ 6x^2 - 4xy - 10y^2} \\ \underline{6x^2 + 6xy\phantom{ - 000}} \\ -10xy - 10y^2 \\ \underline{-10xy - 10y^2} \\ 0 \\ \end{array} \]
Thus, the breadth of the rectangle is \( 3x - 5y \) units.
In simple words: To find the width of a rectangle when you know the area and the length, just divide the area by the length using division.

Exam Tip: Don't forget to write the units ("units") at the end of your final answer, as marks are often deducted for missing units.

Question 4. The area of a rectangular field is \( 25x^2 + 20xy + 3y^2 \) square units. If its length is \( 5x + 3y \) units, find its breadth. Hence, find its perimeter.
Answer:
Given:
Area of the rectangular field = \( 25x^2 + 20xy + 3y^2 \) square units
Length of the field = \( 5x + 3y \) units

We know that:
\( \text{Breadth} = \frac{\text{Area}}{\text{Length}} \)

\( \text{Breadth} = \frac{25x^2 + 20xy + 3y^2}{5x + 3y} \)

Dividing the area by the length using division:
\[ \begin{array}{r} 5x + y \\ 5x + 3y \overline{\smash{)} \ 25x^2 + 20xy + 3y^2} \\ \underline{25x^2 + 15xy\phantom{ + 000}} \\ 5xy + 3y^2 \\ \underline{5xy + 3y^2} \\ 0 \\ \end{array} \]
Thus, the breadth of the rectangular field is \( 5x + y \) units.

Now, we calculate the perimeter:
\( \text{Perimeter} = 2(\text{Length} + \text{Breadth}) \)
\( \implies \text{Perimeter} = 2((5x + 3y) + (5x + y)) \)
\( \implies \text{Perimeter} = 2(10x + 4y) \)
\( \implies \text{Perimeter} = 20x + 8y \) units.
In simple words: First, find the width of the field by dividing the area by the length. Next, add the length and the width together, and multiply the result by two to find the total distance around the field.

Exam Tip: Remember that perimeter is a linear measure, so its unit is "units," not "square units." Make sure you show both steps clearly to get full marks.

Question 5. Divide:
(i) \( 2m^3n^5 \) by \( -mn \)
(ii) \( 5x^2 - 3x \) by \( x \)
(iii) \( 10x^3y - 9xy^2 - 4x^2y^2 \) by \( xy \)
(iv) \( 3y^3 - 9ay^2 - 6ab^2y \) by \( -3y \)
(v) \( x^5 - 15x^4 - 10x^2 \) by \( -5x^2 \)
(vi) \( 12a^2 + ax - 6x^2 \) by \( 3a - 2x \)
(vii) \( 6x^2 - xy - 35y^2 \) by \( 2x - 5y \)
(viii) \( x^3 - 6x^2 + 11x - 6 \) by \( x^2 - 4x + 3 \)
(ix) \( m^3 - 4m^2 + m + 6 \) by \( m^2 - m - 2 \)
Answer:
(i) Dividing the variables:
\( \frac{2m^3n^5}{-mn} = -2m^2n^4 \)

(ii) Separating the terms:
\( \frac{5x^2 - 3x}{x} = \frac{5x^2}{x} - \frac{3x}{x} = 5x - 3 \)

(iii) Splitting the numerator:
\( \frac{10x^3y - 9xy^2 - 4x^2y^2}{xy} = \frac{10x^3y}{xy} - \frac{9xy^2}{xy} - \frac{4x^2y^2}{xy} = 10x^2 - 9y - 4xy \)

(iv) Dividing each term by \( -3y \):
\( \frac{3y^3 - 9ay^2 - 6ab^2y}{-3y} = \frac{3y^3}{-3y} - \frac{9ay^2}{-3y} - \frac{6ab^2y}{-3y} = -y^2 + 3ay + 2ab^2 \)

(v) Performing the division:
\( \frac{x^5 - 15x^4 - 10x^2}{-5x^2} = \frac{x^5}{-5x^2} - \frac{15x^4}{-5x^2} - \frac{10x^2}{-5x^2} = -\frac{1}{5}x^3 + 3x^2 + 2 \)

(vi) Performing polynomial division:
\[ \begin{array}{r} 4a + 3x \\ 3a - 2x \overline{\smash{)} \ 12a^2 + ax - 6x^2} \\ \underline{12a^2 - 8ax\phantom{ + 000}} \\ 9ax - 6x^2 \\ \underline{9ax - 6x^2} \\ 0 \\ \end{array} \]
The quotient is \( 4a + 3x \).

(vii) Performing polynomial division:
\[ \begin{array}{r} 3x + 7y \\ 2x - 5y \overline{\smash{)} \ 6x^2 - xy - 35y^2} \\ \underline{6x^2 - 15xy\phantom{ + 00}} \\ 14xy - 35y^2 \\ \underline{14xy - 35y^2} \\ 0 \\ \end{array} \]
The quotient is \( 3x + 7y \).

(viii) Performing polynomial division:
\[ \begin{array}{r} x - 2 \\ x^2 - 4x + 3 \overline{\smash{)} \ x^3 - 6x^2 + 11x - 6} \\ \underline{x^3 - 4x^2 + 3x\phantom{ - 0}} \\ -2x^2 + 8x - 6 \\ \underline{-2x^2 + 8x - 6} \\ 0 \\ \end{array} \]
The quotient is \( x - 2 \).

(ix) Performing polynomial division:
\[ \begin{array}{r} m - 3 \\ m^2 - m - 2 \overline{\smash{)} \ m^3 - 4m^2 + m + 6} \\ \underline{m^3 - m^2 - 2m\phantom{ + 0}} \\ -3m^2 + 3m + 6 \\ \underline{-3m^2 + 3m + 6} \\ 0 \\ \end{array} \]
The quotient is \( m - 3 \).
In simple words: When dividing by a single term, split and divide each part on top individually. When dividing by an expression with multiple terms, use algebraic long division, finding the matching terms step by step.

Exam Tip: Always make sure to write out all algebraic terms in descending powers of the variable, and place 0 coefficients for any missing power terms to avoid alignment errors during division.

Exercise 11(E)

Simplify:

Question 1. Simplify: \( \frac{x}{2} + \frac{x}{4} \)
Answer:
Find a common denominator, which is 4:
\( \frac{x}{2} + \frac{x}{4} = \frac{2x + x}{4} = \frac{3x}{4} \)
In simple words: To add these fractions, change the first fraction so it has a denominator of 4, then add the numerators together.

Exam Tip: Finding the correct Least Common Multiple (LCM) of the denominators is the key first step to adding fractional expressions successfully.

Question 2. Simplify: \( \frac{a}{10} + \frac{2a}{5} \)
Answer:
Find the LCM of 10 and 5, which is 10:
\( \frac{a}{10} + \frac{2a}{5} = \frac{a + 4a}{10} = \frac{5a}{10} = \frac{a}{2} \)
In simple words: Make the denominators the same by multiplying the top and bottom of the second fraction by 2. Then add them and simplify the final fraction.

Exam Tip: Always reduce your final fractional answer to its simplest terms to ensure you do not lose marks.

Question 3. Simplify: \( \frac{y}{4} + \frac{3y}{5} \)
Answer:
Find the LCM of 4 and 5, which is 20:
\( \frac{y}{4} + \frac{3y}{5} = \frac{5y + 12y}{20} = \frac{17y}{20} \)
In simple words: Change both fractions so they have 20 on the bottom. Multiply the first by 5 and the second by 4, then add them.

Exam Tip: When the denominators have no common factors, their LCM is simply their product.

Question 4. Simplify: \( \frac{x}{2} - \frac{x}{8} \)
Answer:
Find the LCM of 2 and 8, which is 8:
\( \frac{x}{2} - \frac{x}{8} = \frac{4x - x}{8} = \frac{3x}{8} \)
In simple words: Change the first fraction to have a denominator of 8, then subtract the top parts.

Exam Tip: Be careful with subtraction operations and ensure you only subtract the numerators while keeping the common denominator the same.

Question 5. Simplify: \( \frac{3y}{4} - \frac{y}{5} \)
Answer:
Find the LCM of 4 and 5, which is 20:
\( \frac{3y}{4} - \frac{y}{5} = \frac{15y - 4y}{20} = \frac{11y}{20} \)
In simple words: Convert both fractions to a common denominator of 20, then subtract the second numerator from the first.

Exam Tip: Check your final fraction to see if the numerator and denominator share any common factors that can be simplified.

Question 6. Simplify: \( \frac{2p}{3} - \frac{3p}{5} \)
Answer:
To subtract these fractions, find the least common multiple of 3 and 5, which is 15.
\( \frac{2p}{3} - \frac{3p}{5} \)
\( = \frac{5(2p) - 3(3p)}{15} \)
\( = \frac{10p - 9p}{15} \)
\( = \frac{p}{15} \)
In simple words: Find a common bottom number first. For 3 and 5, that number is 15. Change the top numbers to match, then subtract them.

Exam Tip: Always make sure both denominators match before you subtract. Write out the common denominator clearly under a single fraction line.

Question 7. Simplify: \( \frac{k}{2} + \frac{k}{3} + \frac{2k}{5} \)
Answer:
The least common multiple of 2, 3, and 5 is 30. Rewrite each term with the common denominator 30.
\( \frac{k}{2} + \frac{k}{3} + \frac{2k}{5} \)
\( = \frac{15k + 10k + 12k}{30} \)
\( = \frac{37k}{30} \)
In simple words: Find the common bottom number, which is 30. Convert all the top numbers and add them together.

Exam Tip: Double check your addition of the like terms in the numerator to ensure you do not make a simple arithmetic mistake.

Question 8. Simplify: \( \frac{2x}{5} + \frac{3x}{4} - \frac{3x}{5} \)
Answer:
The least common multiple of 5 and 4 is 20. Rewrite each term with 20 as the denominator.
\( \frac{2x}{5} + \frac{3x}{4} - \frac{3x}{5} \)
\( = \frac{4(2x) + 5(3x) - 4(3x)}{20} \)
\( = \frac{8x + 15x - 12x}{20} \)
\( = \frac{23x - 12x}{20} \)
\( = \frac{11x}{20} \)
In simple words: The common bottom number is 20. Convert all three fractions to this denominator, then combine the top values.

Exam Tip: Be careful with the minus sign in the last term. Perform the addition first and then subtract the final term.

Question 9. Simplify: \( \frac{4a}{7} - \frac{2a}{3} + \frac{a}{7} \)
Answer:
The least common multiple of 7 and 3 is 21. Convert the terms to have a common denominator of 21.
\( \frac{4a}{7} - \frac{2a}{3} + \frac{a}{7} \)
\( = \frac{3(4a) - 7(2a) + 3(a)}{21} \)
\( = \frac{12a - 14a + 3a}{21} \)
\( = \frac{15a - 14a}{21} \)
\( = \frac{a}{21} \)
In simple words: Use 21 as the common bottom number. Adjust the top numbers accordingly, then calculate the final sum.

Exam Tip: Group positive terms first before subtracting to make the subtraction step easier and prevent sign errors.

Question 10. Simplify: \( \frac{2b}{5} - \frac{7b}{15} + \frac{13b}{3} \)
Answer:
The least common multiple of 5, 15, and 3 is 15. Express each term with this denominator.
\( \frac{2b}{5} - \frac{7b}{15} + \frac{13b}{3} \)
\( = \frac{3(2b) - 7b + 5(13b)}{15} \)
\( = \frac{6b - 7b + 65b}{15} \)
\( = \frac{71b - 7b}{15} \)
\( = \frac{64b}{15} \)
In simple words: Find a common denominator, which is 15. Rewrite the fractions and combine the numerators.

Exam Tip: Since 15 is already a multiple of 3 and 5, it is the simplest choice for your common denominator.

Question 11. Simplify: \( \frac{6k}{7} - \left(\frac{8k}{9} - \frac{k}{3}\right) \)
Answer:
The least common multiple of 7, 9, and 3 is 63. Solve inside the parentheses first.
\( \frac{6k}{7} - \left(\frac{8k}{9} - \frac{k}{3}\right) \)
\( = \frac{54k - (56k - 21k)}{63} \)
\( = \frac{54k - (35k)}{63} \)
\( = \frac{54k - 35k}{63} \)
\( = \frac{19k}{63} \)
In simple words: First, subtract the terms inside the parentheses. Then, subtract that result from the first fraction using a common denominator of 63.

Exam Tip: Always follow the order of operations by simplifying terms inside parentheses before dealing with the outside sign.

Question 12. Simplify: \( \frac{3a}{8} + \frac{4a}{5} - \left(\frac{a}{2} + \frac{2a}{5}\right) \)
Answer:
The least common multiple of 8, 5, and 2 is 40. Simplify the expression by combining terms.
\( \frac{3a}{8} + \frac{4a}{5} - \left(\frac{a}{2} + \frac{2a}{5}\right) \)
\( = \frac{15a + 32a - (20a + 16a)}{40} \)
\( = \frac{47a - 36a}{40} \)
\( = \frac{11a}{40} \)
In simple words: Find the common bottom number, which is 40. Add the first two terms and subtract the sum of the bracketed terms.

Exam Tip: Be sure to apply the subtraction to the entire sum of the terms inside the brackets.

Question 13. Simplify: \( x + \frac{x}{2} + \frac{x}{3} \)
Answer:
Write \( x \) as \( \frac{x}{1} \). The least common multiple of 1, 2, and 3 is 6.
\( \frac{x}{1} + \frac{x}{2} + \frac{x}{3} \)
\( = \frac{6x + 3x + 2x}{6} \)
\( = \frac{11x}{6} \)
In simple words: Write the whole number x as a fraction over 1. Then use 6 as the common denominator to add them up.

Exam Tip: Remember that any whole number or variable can be written as a fraction with 1 in the denominator.

Question 14. Simplify: \( \frac{y}{5} + y - \frac{19y}{15} \)
Answer:
Write \( y \) as \( \frac{y}{1} \). The least common multiple of 5 and 15 is 15.
\( \frac{y}{5} + \frac{y}{1} - \frac{19y}{15} \)
\( = \frac{3y + 15y - 19y}{15} \)
\( = \frac{18y - 19y}{15} \)
\( = \frac{-y}{15} \)
In simple words: Express y as a fraction over 1. Use 15 as the common denominator, combine the numerators, and simplify.

Exam Tip: When the numerator is negative, write the minus sign clearly in front of the entire fraction.

Question 15. Simplify: \( \frac{x}{5} + \frac{x + 1}{2} \)
Answer:
The least common multiple of 5 and 2 is 10. Multiply each numerator by the corresponding factor.
\( \frac{x}{5} + \frac{x + 1}{2} \)
\( = \frac{2x + 5(x + 1)}{10} \)
\( = \frac{2x + 5x + 5}{10} \)
\( = \frac{7x + 5}{10} \)
In simple words: Find a common bottom number of 10. Multiply the second numerator by 5, then add everything together.

Exam Tip: When multiplying a binomial like \( x + 1 \) by a number, be sure to distribute the number to both terms.

Question 16. Simplify: \( x + \frac{x + 2}{3} \)
Answer:
Express \( x \) as \( \frac{x}{1} \) and use 3 as the common denominator.
\( \frac{x}{1} + \frac{x + 2}{3} \)
\( = \frac{3x + x + 2}{3} \)
\( = \frac{4x + 2}{3} \)
In simple words: Turn x into a fraction over 1. Bring them to the common bottom number 3, and add the top parts.

Exam Tip: Keep the numerator terms separate if they are not like terms - do not add constants to variables.

Question 17. Simplify: \( \frac{3y}{5} - \frac{y + 2}{2} \)
Answer:
The least common multiple of 5 and 2 is 10. Be careful to distribute the negative sign to both terms of the second numerator.
\( \frac{3y}{5} - \frac{y + 2}{2} \)
\( = \frac{2(3y) - 5(y + 2)}{10} \)
\( = \frac{6y - (5y + 10)}{10} \)
\( = \frac{6y - 5y - 10}{10} \)
\( = \frac{y - 10}{10} \)
In simple words: Put the fractions over the common denominator 10. Subtract the second term carefully by changing the signs of both parts.

Exam Tip: A negative sign before a fraction line acts like parentheses - it changes the sign of every term in the numerator.

Question 18. Simplify: \( \frac{2a + 1}{3} + \frac{3a - 1}{2} \)
Answer:
The least common multiple of 3 and 2 is 6. Multiply each numerator by the opposite denominator.
\( \frac{2a + 1}{3} + \frac{3a - 1}{2} \)
\( = \frac{2(2a + 1) + 3(3a - 1)}{6} \)
\( = \frac{4a + 2 + 9a - 3}{6} \)
\( = \frac{13a - 1}{6} \)
In simple words: Find the common denominator 6. Expand the numerators and combine the terms to get the final answer.

Exam Tip: Expand brackets carefully and group the like variable terms and constant terms separately before adding.

Question 19. Simplify: \( \frac{k + 1}{2} + \frac{2k - 1}{3} - \frac{k + 3}{4} \)
Answer:
The least common multiple of 2, 3, and 4 is 12. Write each fraction with this common denominator.
\( \frac{k + 1}{2} + \frac{2k - 1}{3} - \frac{k + 3}{4} \)
\( = \frac{6(k + 1) + 4(2k - 1) - 3(k + 3)}{12} \)
\( = \frac{6k + 6 + 8k - 4 - 3k - 9}{12} \)
\( = \frac{14k - 3k + 6 - 13}{12} \)
\( = \frac{11k - 7}{12} \)
In simple words: Bring everything to the common denominator 12. Distribute the numbers outside each bracket, and simplify the top.

Exam Tip: Watch the negative sign before the third fraction. It must distribute to both \( k \) and \( 3 \).

Question 20. Simplify: \( \frac{m}{5} - \frac{m - 2}{3} + m \)
Answer:
Write \( m \) as \( \frac{m}{1} \). The least common multiple of 5, 3, and 1 is 15.
\( \frac{m}{5} - \frac{m - 2}{3} + \frac{m}{1} \)
\( = \frac{3m - 5(m - 2) + 15m}{15} \)
\( = \frac{3m - 5m + 10 + 15m}{15} \)
\( = \frac{18m - 5m + 10}{15} \)
\( = \frac{13m + 10}{15} \)
In simple words: Express the whole term over 1, then find the common bottom number 15. Solve the top part carefully.

Exam Tip: Be careful when expanding \( -5(m - 2) \). Multiplying two negative numbers yields a positive result, so you get \( +10 \).

Question 21. Simplify: \( \frac{5(x - 4)}{3} + \frac{2(5x - 3)}{5} + \frac{6(x - 4)}{7} \)
Answer:
The least common multiple of 3, 5, and 7 is 105. Rewrite each term over this denominator.
\( \frac{5(x - 4)}{3} + \frac{2(5x - 3)}{5} + \frac{6(x - 4)}{7} \)
\( = \frac{175(x - 4) + 42(5x - 3) + 90(x - 4)}{105} \)
\( = \frac{175x - 700 + 210x - 126 + 90x - 360}{105} \)
\( = \frac{175x + 210x + 90x - 700 - 126 - 360}{105} \)
\( = \frac{475x - 1186}{105} \)
In simple words: Use 105 as the common denominator. Multiply each numerator by the correct factor, expand, and group the like terms.

Exam Tip: Keep your expansion steps separate to prevent simple mistakes when dealing with large numbers.

Question 22. Simplify: \( \left(p + \frac{p}{3}\right)\left(2p + \frac{p}{2}\right)\left(3p - \frac{2p}{3}\right) \)
Answer:
Factor out the variable \( p \) from each set of parentheses and simplify the numerical fractions.
\( \left(p + \frac{p}{3}\right)\left(2p + \frac{p}{2}\right)\left(3p - \frac{2p}{3}\right) \)
\( = p\left(1 + \frac{1}{3}\right) \cdot p\left(2 + \frac{1}{2}\right) \cdot p\left(3 - \frac{2}{3}\right) \)
\( = p^3 \left(\frac{4}{3}\right)\left(\frac{5}{2}\right)\left(\frac{7}{3}\right) \)
\( = p^3 \times \frac{70}{9} \)
\( = \frac{70p^3}{9} \)
In simple words: Simplify each bracket first, then multiply the resulting fractions and variables together.

Exam Tip: Multiplying three \( p \) terms results in \( p^3 \). Do not forget to write the exponent in your final answer.

Question 23. Simplify: \( \frac{7}{30} \text{ of } \left(\frac{p}{3} + \frac{7p}{15}\right) \)
Answer:
First, solve the addition inside the brackets using a common denominator of 15.
\( \frac{7}{30} \text{ of } \left(\frac{p}{3} + \frac{7p}{15}\right) \)
\( = \frac{7}{30} \text{ of } \left(\frac{5p + 7p}{15}\right) \)
\( = \frac{7}{30} \text{ of } \left(\frac{12p}{15}\right) \)
Replace "of" with multiplication and simplify:
\( = \frac{7}{30} \times \frac{12p}{15} \)
\( = \frac{7 \times 4p}{10 \times 15} \)
\( = \frac{14p}{75} \)
In simple words: Add the terms inside the parentheses first. Then, multiply the resulting fraction by \( \frac{7}{30} \).

Exam Tip: Remember that "of" means multiplication. Always complete any operations inside brackets first.

Question 24. Simplify: \( \left(2p + \frac{p}{7}\right) \div \left(\frac{9p}{10} + 4p\right) \)
Answer:
Simplify the expressions inside both sets of parentheses first.
\( \left(2p + \frac{p}{7}\right) \div \left(\frac{9p}{10} + 4p\right) \)
\( = \left(\frac{14p + p}{7}\right) \div \left(\frac{9p + 40p}{10}\right) \)
\( = \frac{15p}{7} \div \frac{49p}{10} \)
To divide, multiply by the reciprocal of the second fraction:
\( = \frac{15p}{7} \times \frac{10}{49p} \)
\( = \frac{150}{343} \)
In simple words: Add the terms in each bracket, turn the division into multiplication by flipping the second fraction, and simplify.

Exam Tip: The variable \( p \) appears in both the numerator and denominator, so they cancel out completely.

Question 25. Simplify: \( \left(\frac{5k}{8} - \frac{3k}{5}\right) \div \frac{k}{4} \)
Answer:
Find a common denominator of 40 to subtract the terms inside the parentheses.
\( \left(\frac{5k}{8} - \frac{3k}{5}\right) \div \frac{k}{4} \)
\( = \left(\frac{25k - 24k}{40}\right) \div \frac{k}{4} \)
\( = \frac{k}{40} \div \frac{k}{4} \)
Multiply by the reciprocal of the divisor:
\( = \frac{k}{40} \times \frac{4}{k} \)
\( = \frac{1}{10} \)
In simple words: Subtract the fractions in the bracket first. Then, flip the second fraction and multiply.

Exam Tip: Do not forget to cancel the variable \( k \) in the final multiplication step to get a constant numerical value.

Question 26. Simplify: \( \left(\frac{y}{6} + \frac{2y}{3}\right) \div \left(y + \frac{2y - 1}{3}\right) \)
Answer:
Simplify the expressions inside both sets of parentheses.
\( \left(\frac{y}{6} + \frac{2y}{3}\right) \div \left(y + \frac{2y - 1}{3}\right) \)
\( = \left(\frac{y + 4y}{6}\right) \div \left(\frac{3y + 2y - 1}{3}\right) \)
\( = \frac{5y}{6} \div \frac{5y - 1}{3} \)
Multiply by the reciprocal of the second expression:
\( = \frac{5y}{6} \times \frac{3}{5y - 1} \)
\( = \frac{5y}{2(5y - 1)} \)
\( = \frac{5y}{10y - 2} \)
In simple words: Simplify both parts first. Then divide by multiplying the first part by the reciprocal of the second part.

Exam Tip: You can leave the denominator in factored form or expand it; both are correct. Expanding to \( 10y - 2 \) is generally preferred.

Exercise 11(F)

Enclose the given terms in brackets as required:

Question 1. \( x - y - z = x - (\text{.......}) \)
Answer:
\( x - y - z = x - (y + z) \)
In simple words: When you put a negative sign outside a bracket, change all the signs of the terms inside.

Exam Tip: Always verify by expanding the bracket to see if you get back the original expression.

Question 2. \( x^2 - xy^2 - 2xy - y^2 = x^2 - (\text{.......}) \)
Answer:
\( x^2 - xy^2 - 2xy - y^2 = x^2 - (xy^2 + 2xy + y^2) \)
In simple words: Since we grouped all terms after the first under a negative sign, we changed each of their signs to positive.

Exam Tip: A negative sign outside parentheses acts like multiplying each term inside by \( -1 \).

Question 3. \( 4a - 9 + 2b - 6 = 4a - (\text{.......}) \)
Answer:
\( 4a - 9 + 2b - 6 = 4a - (9 - 2b + 6) \)
In simple words: To group these terms inside brackets with a negative sign outside, reverse the sign of each term.

Exam Tip: Make sure you carefully flip both negative and positive signs when enclosing terms inside parentheses with a leading negative sign.

Question 4. \( x^2 - y^2 + z^2 + 3x - 2y = x^2 - (\text{.......}) \)
Answer:
\( x^2 - y^2 + z^2 + 3x - 2y = x^2 - (y^2 - z^2 - 3x + 2y) \)
In simple words: To group the terms behind the minus sign, change their signs to their exact opposites inside the bracket.

Exam Tip: Grouping terms under a negative sign requires reversing the signs of all enclosed terms to maintain algebraic equality.

Question 5. \( -2a^2 + 4ab - 6a^2b^2 + 8ab^2 = -2a (\text{.......}) \)
Answer:
\( -2a^2 + 4ab - 6a^2b^2 + 8ab^2 = -2a(a - 2b + 3ab^2 - 4b^2) \)
In simple words: Factor out \( -2a \) from every term, which changes their signs and divides each coefficient and variable power by \( -2a \).

Exam Tip: Ensure that when factoring out a negative common monomial, you change the sign of each term inside the bracket.

Simplify:

Question 6. Simplify: \( 2x - (x + 2y - z) \)
Answer:
Expand the parentheses by distributing the negative sign:
\( 2x - (x + 2y - z) \)
\( = 2x - x - 2y + z \)
Combine the like terms:
\( = x - 2y + z \)
In simple words: Remove the brackets by changing the signs of the terms inside, then combine the \( x \) terms.

Exam Tip: Be careful to apply the negative sign to all terms inside the parentheses, including the last term.

Question 7. Simplify: \( p + q - (p - q) + (2p - 3q) \)
Answer:
Remove the brackets carefully, changing signs where a negative is outside:
\( p + q - (p - q) + (2p - 3q) \)
\( = p + q - p + q + 2p - 3q \)
Group like terms:
\( = (p - p + 2p) + (q + q - 3q) \)
\( = 2p - q \)
In simple words: Expand the brackets. The minus sign changes the terms inside the first bracket. Then group the \( p \) and \( q \) terms.

Exam Tip: Grouping similar terms together before adding or subtracting them helps to keep your work organized and accurate.

Question 8. Simplify: \( 9x - (-4x + 5) \)
Answer:
Expand the brackets by distributing the negative sign:
\( 9x - (-4x + 5) \)
\( = 9x + 4x - 5 \)
Combine the like terms:
\( = 13x - 5 \)
In simple words: The minus outside changes the signs inside the bracket, turning \( -4x \) into \( +4x \) and \( +5 \) into \( -5 \).

Exam Tip: Remember that "minus of minus" makes a plus, so \( -(-4x) \) becomes \( +4x \).

Question 9. Simplify: \( 6a - (-5a - 8b) + (3a + b) \)
Answer:
Expand both sets of brackets:
\( 6a - (-5a - 8b) + (3a + b) \)
\( = 6a + 5a + 8b + 3a + b \)
Group and combine the like terms:
\( = (6a + 5a + 3a) + (8b + b) \)
\( = 14a + 9b \)
In simple words: Open the brackets first. Make sure to flip the signs inside the first bracket because of the minus sign outside.

Exam Tip: A plus sign in front of a bracket does not change any signs inside, so \( +(3a + b) \) simply remains \( 3a + b \).

Question 10. Simplify: \( (p - 2q) - (3q - r) \)
Answer:
Open the brackets, changing the signs for the second bracket:
\( (p - 2q) - (3q - r) \)
\( = p - 2q - 3q + r \)
Combine the \( q \) terms:
\( = p - 5q + r \)
In simple words: Expand the terms. The minus outside changes the second bracket, then combine the similar terms.

Exam Tip: Keep the non-like terms \( p \) and \( r \) separate in your final simplified expression.

Question 11. Simplify: \( 9a(2b - 3a + 7c) \)
Answer:
Distribute \( 9a \) to each term inside the parentheses:
\( 9a(2b - 3a + 7c) \)
\( = (9a \times 2b) - (9a \times 3a) + (9a \times 7c) \)
\( = 18ab - 27a^2 + 63ca \)
In simple words: Multiply the term outside, 9a, by every single term inside the bracket.

Exam Tip: Remember that \( a \times a = a^2 \). Make sure to write the exponent correctly when multiplying the variables.

Question 12. Simplify: \( -5m(-2m + 3n - 7p) \)
Answer:
Multiply each term inside the bracket by \( -5m \):
\( -5m(-2m + 3n - 7p) \)
\( = (-5m \times -2m) + (-5m \times 3n) - (-5m \times 7p) \)
\( = 10m^2 - 15mn + 35mp \)
In simple words: Multiply \( -5m \) with each term inside. Watch out for negative numbers multiplying each other to make a positive.

Exam Tip: Be extra careful with signs when distributing a negative coefficient like \( -5m \).

Question 13. Simplify: \( -2x(x + y) + x^2 \)
Answer:
Distribute \( -2x \) to the terms inside the parentheses first:
\( -2x(x + y) + x^2 \)
\( = -2x^2 - 2xy + x^2 \)
Combine the like terms:
\( = -x^2 - 2xy \)
In simple words: Multiply \( -2x \) through the bracket, then add \( x^2 \) and combine the similar terms.

Exam Tip: Group \( -2x^2 \) and \( x^2 \) together first. Remember that \( -2x^2 + x^2 = -x^2 \).

Question 14. Simplify: \( b\left(2b - \frac{1}{b}\right) - 2b\left(b - \frac{1}{b}\right) \)
Answer:
Distribute each term outside the brackets to the terms inside:
\( b\left(2b - \frac{1}{b}\right) - 2b\left(b - \frac{1}{b}\right) \)
\( = (b \times 2b) - \left(b \times \frac{1}{b}\right) - (2b \times b) + \left(2b \times \frac{1}{b}\right) \)
\( = 2b^2 - 1 - 2b^2 + 2 \)
Group and combine the terms:
\( = (2b^2 - 2b^2) + (2 - 1) \)
\( = 1 \)
In simple words: Expand both parts. The variable \( b \) in the numerator and denominator will cancel out. Combine what is left.

Exam Tip: Recall that \( b \times \frac{1}{b} = 1 \). Keep track of your signs as you distribute the negative term.

Question 15. Simplify: \( 8(2a + 3b - c) - 10(a + 2b + 3c) \)
Answer:
Distribute 8 and \( -10 \) to their respective brackets:
\( 8(2a + 3b - c) - 10(a + 2b + 3c) \)
\( = 16a + 24b - 8c - 10a - 20b - 30c \)
Group and combine the like terms:
\( = (16a - 10a) + (24b - 20b) - (8c + 30c) \)
\( = 6a + 4b - 38c \)
In simple words: Multiply everything inside the first bracket by 8, and everything in the second bracket by -10. Then put like terms together.

Exam Tip: Be very careful when distributing \( -10 \) across the second bracket. All signs inside will change.

Question 16. Simplify: \( a\left(a + \frac{1}{a}\right) - b\left(b - \frac{1}{b}\right) - c\left(c + \frac{1}{c}\right) \)
Answer:
Distribute each factor outside to its corresponding bracket:
\( a\left(a + \frac{1}{a}\right) - b\left(b - \frac{1}{b}\right) - c\left(c + \frac{1}{c}\right) \)
\( = a^2 + 1 - b^2 + 1 - c^2 - 1 \)
Combine the constant terms:
\( = a^2 - b^2 - c^2 + 1 \)
In simple words: Multiply each term. The variables cancel out to become 1, -1, or +1. Combine the numbers at the end.

Exam Tip: Watch the signs: \( -b \times -\frac{1}{b} = +1 \), and \( -c \times +\frac{1}{c} = -1 \).

Question 17. Simplify: \( 5x(2x + 3y) - 2x(x - 9y) \)
Answer:
Distribute \( 5x \) and \( -2x \) to expand the brackets:
\( 5x(2x + 3y) - 2x(x - 9y) \)
\( = 10x^2 + 15xy - 2x^2 + 18xy \)
Combine the like terms:
\( = (10x^2 - 2x^2) + (15xy + 18xy) \)
\( = 8x^2 + 33xy \)
In simple words: Multiply both brackets, make sure to change the signs when distributing the negative term, then combine the similar parts.

Exam Tip: Grouping terms like \( x^2 \) and \( xy \) separately ensures you do not accidentally combine unlike terms.

Question 18. Simplify: \( a + (b + c - d) \)
Answer:
Since there is a positive sign outside the bracket, the signs inside remain the same:
\( a + (b + c - d) \)
\( = a + b + c - d \)
In simple words: A plus sign in front of parentheses means you can simply remove them without changing any signs.

Exam Tip: Parentheses preceded by a plus sign can be removed directly, preserving all internal signs exactly as they are.

Question 19. Simplify: \( 5 - 8x - 6 - x \)
Answer:
Group the like terms (constants and variables) together:
\( 5 - 8x - 6 - x \)
\( = (5 - 6) - (8x + x) \)
\( = -1 - 9x \)
In simple words: Put the regular numbers together and the x numbers together, then add or subtract them.

Exam Tip: Remember that a lone variable like \( -x \) has an implicit coefficient of 1, meaning it is \( -1x \).

Question 20. Simplify: \( 2a + (b - \overline{a - b}) \)
Answer:
First, resolve the vinculum bar over \( a - b \):
\( 2a + (b - \overline{a - b}) \)
\( = 2a + (b - (a - b)) \)
\( = 2a + (b - a + b) \)
\( = 2a + (2b - a) \)
Now open the parentheses:
\( = 2a + 2b - a \)
\( = a + 2b \)
In simple words: Treat the bar like a bracket. Simplify that part first, then simplify the parentheses, and finally combine similar terms.

Exam Tip: A bar bracket (vinculum) has high priority. Always treat it as a set of parentheses with a minus sign in front of it if there is one.

Question 21. Simplify: \( 3x + [4x - (6x - 3)] \)
Answer:
First simplify inside the parentheses, then the square brackets:
\( 3x + [4x - (6x - 3)] \)
\( = 3x + [4x - 6x + 3] \)
\( = 3x + [-2x + 3] \)
\( = 3x - 2x + 3 \)
\( = x + 3 \)
In simple words: Work from the inside out. First, open the inner parentheses, then simplify the square brackets, and combine terms.

Exam Tip: Working systematically from the innermost brackets outward is the safest way to avoid sign errors.

Question 22. Simplify: \( 5b - \{6a + (8 - b - a)\} \)
Answer:
First, remove the parentheses since they are preceded by a plus sign:
\( 5b - \{6a + (8 - b - a)\} \)
\( = 5b - \{6a + 8 - b - a\} \)
Combine like terms inside the braces:
\( = 5b - \{5a - b + 8\} \)
Now distribute the negative sign to remove the braces:
\( = 5b - 5a + b - 8 \)
Combine the remaining like terms:
\( = -5a + 6b - 8 \)
In simple words: Open the inside bracket first, combine the a terms, then open the outer bracket by changing all signs inside, and simplify.

Exam Tip: Be sure to distribute the minus sign to every single term inside the curly braces when opening them.

Question 23. Simplify: \( 2x - [5y - (3x - y) + x] \)
Answer:
Start by opening the innermost parentheses:
\( 2x - [5y - (3x - y) + x] \)
\( = 2x - [5y - 3x + y + x] \)
Simplify inside the square brackets:
\( = 2x - [6y - 2x] \)
Now open the square brackets:
\( = 2x - 6y + 2x \)
Combine the like terms:
\( = 4x - 6y \)
In simple words: Open the inside brackets and swap signs. Combine like terms, then open the square brackets and simplify.

Exam Tip: Simplify the terms inside the brackets as much as possible before removing the outer bracket to keep steps clean.

Question 24. Simplify: \( 6a - 3(a + b - 2) \)
Answer:
Distribute the \( -3 \) to the terms inside the parentheses:
\( 6a - 3(a + b - 2) \)
\( = 6a - 3a - 3b + 6 \)
Combine like terms:
\( = 3a - 3b + 6 \)
In simple words: Multiply \( -3 \) with everything inside the bracket, then subtract \( 3a \) from \( 6a \).

Exam Tip: Remember to multiply \( -3 \) by \( -2 \) to get a positive \( +6 \) at the end.

Question 25. Simplify: \( 8[m + 2n - p - 7(2m - n + 3p)] \)
Answer:
First distribute \( -7 \) inside the square brackets:
\( 8[m + 2n - p - 7(2m - n + 3p)] \)
\( = 8[m + 2n - p - 14m + 7n - 21p] \)
Combine like terms inside the brackets:
\( = 8[-13m + 9n - 22p] \)
Distribute 8 to all terms:
\( = -104m + 72n - 176p \)
In simple words: Multiply the inner term by \( -7 \), collect similar variables together, and finally multiply everything by 8.

Exam Tip: Follow the correct hierarchy: simplify the inner multiplication first before multiplying by the outer coefficient.

Question 26. Simplify: \( \{9 - (4p - 6q)\} - \{3q - (5p - 10)\} \)
Answer:
Open the inner parentheses first:
\( \{9 - (4p - 6q)\} - \{3q - (5p - 10)\} \)
\( = \{9 - 4p + 6q\} - \{3q - 5p + 10\} \)
Now distribute the negative sign before the second set of braces:
\( = 9 - 4p + 6q - 3q + 5p - 10 \)
Group and combine like terms:
\( = (-4p + 5p) + (6q - 3q) + (9 - 10) \)
\( = p + 3q - 1 \)
In simple words: Open the inside brackets first, change signs where needed, open the braces, and combine similar variables and numbers.

Exam Tip: Treat curly braces just like normal brackets; they follow the same sign distribution rules.

Question 27. Simplify: \( 2[a - 3\{a + 5(a - 2) + 7\}] \)
Answer:
Start simplifying from the innermost bracket:
\( 2[a - 3\{a + 5(a - 2) + 7\}] \)
\( = 2[a - 3\{a + 5a - 10 + 7\}] \)
Combine like terms inside the braces:
\( = 2[a - 3\{6a - 3\}] \)
Distribute \( -3 \) across the terms inside the braces:
\( = 2[a - 18a + 9] \)
Combine like terms inside the square brackets:
\( = 2[-17a + 9] \)
Now multiply by 2:
\( = -34a + 18 \)
In simple words: Work your way from the inside out. Open parentheses, combine terms, open braces, combine terms, open brackets, and multiply.

Exam Tip: Be sure to perform each step systematically to avoid missing any term during multiple expansions.

Question 28. Simplify: \( 5a - [6a - \{9a - (10a - \overline{4a - 3a})\}] \)
Answer:
First, resolve the vinculum bar over \( 4a - 3a \):
\( 5a - [6a - \{9a - (10a - \overline{4a - 3a})\}] \)
\( = 5a - [6a - \{9a - (10a - (4a - 3a))\}] \)
\( = 5a - [6a - \{9a - (10a - a)\}] \)
\( = 5a - [6a - \{9a - 9a\}] \)
\( = 5a - [6a - 0] \)
\( = 5a - 6a \)
\( = -a \)
In simple words: Simplify the part under the bar first. Then work through the parentheses, the braces, and the square brackets.

Exam Tip: The vinculum is processed before parentheses. Simplifying inside brackets early often collapses the entire expression quickly.

Question 29. Simplify: \( 9x + 5 - [4x - \{3x - 2(4x - 3)\}] \)
Answer:
Simplify from the innermost parentheses:
\( 9x + 5 - [4x - \{3x - 2(4x - 3)\}] \)
\( = 9x + 5 - [4x - \{3x - 8x + 6\}] \)
Combine like terms inside the braces:
\( = 9x + 5 - [4x - \{-5x + 6\}] \)
Open the braces by distributing the negative sign:
\( = 9x + 5 - [4x + 5x - 6] \)
Simplify inside the square brackets:
\( = 9x + 5 - [9x - 6] \)
Now open the square brackets:
\( = 9x + 5 - 9x + 6 \)
Combine the terms:
\( = 11 \)
In simple words: Distribute the 2, combine the x terms inside, open each outer set of brackets, and combine terms until only a number is left.

Exam Tip: Notice that the \( x \) terms cancel out entirely, leaving a constant numerical value as the final answer.

Question 30. Simplify: \( (x + y - z)x + (z + x - y)y - (x + y - z)z \)
Answer:
Multiply each term inside the brackets by the factor outside:
\( (x + y - z)x + (z + x - y)y - (x + y - z)z \)
\( = (x^2 + xy - zx) + (zy + xy - y^2) - (zx + zy - z^2) \)
Open the brackets carefully:
\( = x^2 + xy - zx + zy + xy - y^2 - zx - zy + z^2 \)
Group and combine the like terms:
\( = x^2 - y^2 + z^2 + (xy + xy) + (zy - zy) + (-zx - zx) \)
\( = x^2 - y^2 + z^2 + 2xy - 2zx \)
In simple words: Expand each bracket, group like terms, and cancel opposite terms to get the simplified expression.

Exam Tip: Be methodical when grouping terms; cross out terms lightly on your scratch paper once you have combined them.

Question 31. Simplify: \( -1[a - 3\{b - 4(a - b - 8) + 4a\} + 10] \)
Answer:
Start simplifying from the innermost bracket:
\( -1[a - 3\{b - 4(a - b - 8) + 4a\} + 10] \)
\( = -1[a - 3\{b - 4a + 4b + 32 + 4a\} + 10] \)
Simplify inside the braces by combining terms:
\( = -1[a - 3\{5b + 32\} + 10] \)
Now distribute the \( -3 \):
\( = -1[a - 15b - 96 + 10] \)
Simplify inside the square brackets:
\( = -1[a - 15b - 86] \)
Finally, multiply the entire expression by \( -1 \):
\( = -a + 15b + 86 \)
In simple words: Open the innermost brackets, combine similar parts, distribute the outer factors step by step, and multiply by -1 at the end.

Exam Tip: Always make sure to carry through the sign of every single term when expanding nested brackets.

Question 32. Simplify: \( p^2 - [x^2 - \{x^2 - (q^2 - \overline{x^2 - q^2}) - 2y^2\}] \)
Answer:
Simplify the vinculum bar first:
\( p^2 - [x^2 - \{x^2 - (q^2 - \overline{x^2 - q^2}) - 2y^2\}] \)
\( = p^2 - [x^2 - \{x^2 - (q^2 - (x^2 - q^2)) - 2y^2\}] \)
\( = p^2 - [x^2 - \{x^2 - (q^2 - x^2 + q^2) - 2y^2\}] \)
\( = p^2 - [x^2 - \{x^2 - (2q^2 - x^2) - 2y^2\}] \)
Now open the parentheses inside the braces:
\( = p^2 - [x^2 - \{x^2 - 2q^2 + x^2 - 2y^2\}] \)
\( = p^2 - [x^2 - \{2x^2 - 2q^2 - 2y^2\}] \)
Open the braces:
\( = p^2 - [x^2 - 2x^2 + 2q^2 + 2y^2] \)
\( = p^2 - [-x^2 + 2q^2 + 2y^2] \)
Finally, open the square brackets:
\( = p^2 + x^2 - 2q^2 - 2y^2 \)
In simple words: Treat the bar like a bracket and resolve it first. Work your way outward, opening one set of brackets at a time, changing signs appropriately.

Exam Tip: Work through nested expressions from the inside out to avoid any sign errors with double negatives.

Question 33. Simplify: \( 10 - \{4a - (7 - \overline{a - 5}) - (5a - \overline{1 + a})\} \)
Answer:
First, resolve both vinculum bars:
\( 10 - \{4a - (7 - \overline{a - 5}) - (5a - \overline{1 + a})\} \)
\( = 10 - \{4a - (7 - (a - 5)) - (5a - (1 + a))\} \)
Simplify inside the parentheses:
\( = 10 - \{4a - (7 - a + 5) - (5a - 1 - a)\} \)
\( = 10 - \{4a - (12 - a) - (4a - 1)\} \)
Now expand the parentheses inside the braces:
\( = 10 - \{4a - 12 + a - 4a + 1\} \)
Simplify inside the braces:
\( = 10 - \{a - 11\} \)
Open the braces:
\( = 10 - a + 11 \)
\( = 21 - a \)
In simple words: Resolve the bar brackets first, simplify each group inside the parentheses, expand everything, and combine the remaining numbers.

Exam Tip: Be careful with the minus sign before the parentheses. It must distribute to all terms within that group.

Question 34. Simplify: \( 7a - [8a - \{11a - (12a - \overline{6a - 5a})\}] \)
Answer:
Start with the vinculum bar inside the parentheses:
\( 7a - [8a - \{11a - (12a - \overline{6a - 5a})\}] \)
\( = 7a - [8a - \{11a - (12a - a)\}] \)
Simplify inside the parentheses:
\( = 7a - [8a - \{11a - 11a\}] \)
Simplify inside the braces:
\( = 7a - [8a - 0] \)
\( = 7a - 8a \)
\( = -a \)
In simple words: Simplify the part under the bar first. Solve the parentheses, then the braces, and finally the square brackets.

Exam Tip: Look for opportunities to simplify terms early; here, the terms inside the braces subtract to zero, which greatly simplifies the problem.

Question 35. Simplify: \( 8x - [4y - \{4x + (2x - \overline{2y - 2x})\}] \)
Answer:
Resolve the vinculum bar inside the parentheses:
\( 8x - [4y - \{4x + (2x - \overline{2y - 2x})\}] \)
\( = 8x - [4y - \{4x + (2x - (2y - 2x))\}] \)
\( = 8x - [4y - \{4x + (2x - 2y + 2x)\}] \)
Simplify inside the parentheses:
\( = 8x - [4y - \{4x + (4x - 2y)\}] \)
Open the parentheses and simplify inside the braces:
\( = 8x - [4y - \{4x + 4x - 2y\}] \)
\( = 8x - [4y - \{8x - 2y\}] \)
Open the braces:
\( = 8x - [4y - 8x + 2y] \)
Simplify inside the square brackets:
\( = 8x - [6y - 8x] \)
Open the square brackets:
\( = 8x - 6y + 8x \)
Combine the terms:
\( = 16x - 6y \)
In simple words: Remove the bar first, then work from parentheses to braces, and finally to square brackets, changing signs when opening.

Exam Tip: Be sure to combine the like terms at each stage of bracket simplification to keep the algebra manageable.

Question 36. Simplify: \( x - \left(3y - \overline{4z - 3x} + 2z - \overline{5y - 7x}\right) \)
Answer:
Start by resolving the vinculum bars inside the parentheses:
\( x - \left(3y - \overline{4z - 3x} + 2z - \overline{5y - 7x}\right) \)
\( = x - \left(3y - (4z - 3x) + 2z - (5y - 7x)\right) \)
Open the inner brackets inside the parentheses:
\( = x - (3y - 4z + 3x + 2z - 5y + 7x) \)
Group and combine like terms inside the parentheses:
\( = x - ((3x + 7x) + (3y - 5y) + (-4z + 2z)) \)
\( = x - (10x - 2y - 2z) \)
Now distribute the negative sign outside the parentheses:
\( = x - 10x + 2y + 2z \)
Combine the remaining like terms:
\( = -9x + 2y + 2z \)
In simple words: Simplify the parts under the bars first, open the inside brackets, group the similar terms, and then subtract the whole group from the first term.

Exam Tip: Watch the signs carefully when removing the final outer parentheses; every term inside must change sign.