Selina Concise Solutions for ICSE Class 6 Mathematics Chapter 19 Fundamental Operations

Question 1. Fill in the blanks:
(i) 5 + 4 = ………… and \( 5x + 4x \) = ………….
(ii) 12 + 18 = ………… and \( 12x^2y + 18x^2y \) = ………….
(iii) 7 + 16 = ………….. and \( 7a + 16b \) = …………
(iv) 1 + 3 = ………… and \( x^2y + 3xy^2 \) = ………..
(v) 7 – 4 = …………… and \( 7ab – 4ab \) = …………..
(vi) 12 – 5 = ………… and \( 12x – 5y \) = ……………
(vii) 35 – 16 = ………….. and \( 35ab – 16ba \) = ………….
(viii) 28 – 13 = …………. and \( 28ax^2 – 13a^2x \) = ………….
Answer:
(i) \( 5 + 4 = 9 \) and \( 5x + 4x = 9x \)
(ii) \( 12 + 18 = 30 \) and \( 12x^2y + 18x^2y = 30x^2y \)
(iii) \( 7 + 16 = 23 \) and \( 7a + 16b = 7a + 16b \)
(iv) \( 1 + 3 = 4 \) and \( x^2y + 3xy^2 = x^2y + 3xy^2 \)
(v) \( 7 - 4 = 3 \) and \( 7ab - 4ab = 3ab \)
(vi) \( 12 - 5 = 7 \) and \( 12x - 5y = 12x - 5y \)
(vii) \( 35 - 16 = 19 \) and \( 35ab - 16ba = 19ab \)
(viii) \( 28 - 13 = 15 \) and \( 28ax^2 - 13a^2x = 28ax^2 - 13a^2x \)
In simple words: When adding or subtracting, we can only combine "like terms" that have the exact same variables and powers. Unlike terms cannot be merged together.

Exam Tip: Be careful with variables like \( ba \) and \( ab \) - they are like terms because the order of multiplication does not change their value. However, terms with different powers, such as \( ax^2 \) and \( a^2x \), are unlike terms and cannot be combined.

Question 2. Fill in the blanks:
(i) The sum of – 2 and – 5 = …………. and the sum of – 2x and – 5x = …………….
(ii) The sum of 8 and – 3 = ………….. and the sum of 8ab and – 3ab = ………….
(iii) The sum of – 15 and – 4 = …………….. and the sum of – 15x and -4y = ………………
(iv) 15 + 8 + 3 = ……….. and 15x + 8y + 3x = …………….
(v) 12 – 9 + 15 = …………… and 12ab – 9ab + 15ba = ……………..
(vi) 25 – 7 – 9 = and 25xy – 7xy – 9yx = ……………
(vii) – 4 – 6 – 5 = …………. and – 4ax – 6ax – 5ay = …………….
Answer:
(i) The total of \( -2 \) and \( -5 \) is \( -7 \), and the total of \( -2x \) and \( -5x \) is \( -7x \).
(ii) The total of \( 8 \) and \( -3 \) is \( 5 \), and the total of \( 8ab \) and \( -3ab \) is \( 5ab \).
(iii) The total of \( -15 \) and \( -4 \) is \( -19 \), and the total of \( -15x \) and \( -4y \) is \( -15x - 4y \).
(iv) \( 15 + 8 + 3 = 26 \) and \( 15x + 8y + 3x = 18x + 8y \).
(v) \( 12 - 9 + 15 = 18 \) and \( 12ab - 9ab + 15ba = 18ab \).
(vi) \( 25 - 7 - 9 = 9 \) and \( 25xy - 7xy - 9yx = 9xy \).
(vii) \( -4 - 6 - 5 = -15 \) and \( -4ax - 6ax - 5ay = -10ax - 5ay \).
In simple words: When combining signed numbers, same-sign numbers add up and keep their sign. For variables, only add or subtract matching variable parts, leaving others separate.

Exam Tip: Pay close attention to signs. Adding negative numbers makes the result more negative, and always keep unlike terms separated in your final answer.

Question 3. Add:
(i) \( 8xy \) and \( 3xy \)
(ii) \( 2xyz \), \( xyz \) and \( 6xyz \)
(iii) \( 2a \), \( 3a \) and \( 4b \)
(iv) \( 3x \) and \( 2y \)
(v) \( 5m \), \( 3n \) and \( 4p \)
(vi) \( 6a \), \( 3a \) and \( 9ab \)
(vii) \( 3p \), \( 4q \) and \( 9q \)
(viii) \( 5ab \), \( 4ba \) and \( 6b \)
(ix) \( 50pq \), \( 30pq \) and \( 10pr \)
(x) \( -2y \), \( -y \) and \( -3y \)
(xi) \( -3b \) and \( -b \)
(xii) \( 5b \), \( -4b \) and \( -10b \)
(xiii) \( -2c \), \( -c \) and \( -5c \)
Answer:
(i) \( 8xy + 3xy = 11xy \)
(ii) \( 2xyz + xyz + 6xyz = (2 + 1 + 6)xyz = 9xyz \)
(iii) \( 2a + 3a + 4b = (2 + 3)a + 4b = 5a + 4b \)
(iv) \( 3x + 2y = 3x + 2y \)
(v) \( 5m + 3n + 4p = 5m + 3n + 4p \)
(vi) \( 6a + 3a + 9ab = (6 + 3)a + 9ab = 9a + 9ab \)
(vii) \( 3p + 4q + 9q = 3p + (4 + 9)q = 3p + 13q \)
(viii) \( 5ab + 4ba + 6b = (5 + 4)ab + 6b = 9ab + 6b \)
(ix) \( 50pq + 30pq + 10pr = (50 + 30)pq + 10pr = 80pq + 10pr \)
(x) \( (-2y) + (-y) + (-3y) = -(2 + 1 + 3)y = -6y \)
(xi) \( (-3b) + (-b) = -(3 + 1)b = -4b \)
(xii) \( 5b + (-4b) + (-10b) = 5b - (4 + 10)b = 5b - 14b = -9b \)
(xiii) \( (-2c) + (-c) + (-5c) = -(2 + 1 + 5)c = -8c \)
In simple words: When we add terms together, we sum the coefficients of the matching terms. If the terms have different variables, we just write them next to each other with a plus sign.

Exam Tip: Remember that a lone variable like \( xyz \) has an implicit coefficient of 1. Do not forget to count it when adding like terms.

Question 4. Evaluate:
(i) \( 6a - a - 5a - 2a \)
(ii) \( 2b - 3b - b + 4b \)
(iii) \( 3x - 2x - 4x + 7x \)
(iv) \( 5ab + 2ab - 6ab + ab \)
(v) \( 8x - 5y - 3x + 10y \)
Answer:
(i) \( 6a - a - 5a - 2a = 6a - (1 + 5 + 2)a \)
\( \implies 6a - 8a = -2a \)
(ii) \( 2b - 3b - b + 4b = 2b + 4b - (3 + 1)b \)
\( \implies 6b - 4b = 2b \)
(iii) \( 3x - 2x - 4x + 7x = 3x + 7x - 2x - 4x \)
\( \implies (3 + 7)x - (2 + 4)x \)
\( \implies 10x - 6x = 4x \)
(iv) \( 5ab + 2ab - 6ab + ab = 5ab + 2ab + ab - 6ab \)
\( \implies 8ab - 6ab = 2ab \)
(v) \( 8x - 5y - 3x + 10y = 8x - 3x + 10y - 5y \)
\( \implies 5x + 5y \)
In simple words: Group positive and negative terms of the same type together first. This makes it much easier to add and subtract their coefficients.

Exam Tip: Be careful with signs when grouping terms. Factoring out the variable can help prevent mistakes during addition and subtraction.

Question 5. Evaluate:
(i) \( -7x + 9x + 2x - 2x \)
(ii) \( 5ab - 2ab - 8ab + 6ab \)
(iii) \( -8a - 3a + 12a + 13a - 6a \)
(iv) \( 19abc - 11abc - 12abc + 14abc \)
Answer:
(i) \( -7x + 9x + 2x - 2x = 9x + 2x - 7x - 2x \)
\( \implies 11x - 9x = 2x \)
(ii) \( 5ab - 2ab - 8ab + 6ab = 5ab + 6ab - 2ab - 8ab \)
\( \implies 11ab - 10ab = ab \)
(iii) \( -8a - 3a + 12a + 13a - 6a = 12a + 13a - (8a + 3a + 6a) \)
\( \implies 25a - 17a = 8a \)
(iv) \( 19abc - 11abc - 12abc + 14abc = abc(19 - 11 - 12 + 14) \)
\( \implies abc(33 - 23) = 10abc \)
In simple words: Collect all positive coefficients and add them up, then collect all negative coefficients and add them up. Finally, find the difference between the two sums.

Exam Tip: Grouping terms with positive signs first and negative signs second is a systematic way to solve longer expressions without making arithmetic errors.

Question 6. Subtract the first term from the second:
(i) \( 4ab, 6ba \)
(ii) \( 4.8b, 6.8b \)
(iii) \( 3.5abc, 10.5abc \)
(iv) \( 3\frac{1}{2}mn, 8\frac{1}{2}nm \)
Answer:
(i) \( 6ba - 4ab = 2ab \) (since \( ba = ab \))
(ii) \( 6.8b - 4.8b = 2b \)
(iii) \( 10.5abc - 3.5abc = 7abc \)
(iv) \( 8\frac{1}{2}nm - 3\frac{1}{2}nm = \frac{17}{2}mn - \frac{7}{2}mn \)
\( \implies \frac{17mn - 7mn}{2} = \frac{10mn}{2} = 5mn \)
In simple words: Since we need to subtract the first item from the second, we write the second item first, then write a minus sign followed by the first item, and subtract like usual.

Exam Tip: Pay special attention to the phrasing "subtract A from B," which means you must calculate \( B - A \), not \( A - B \).

Question 7. Simplify:
(i) \( 2a^2b^2 + 5ab^2 + 8a^2b^2 - 3ab^2 \)
(ii) \( 4a + 3b - 2a - b \)
(iii) \( 2xy + 4yz + 5xy + 3yz - 6xy \)
(iv) \( ab + 15ab - 11ab - 2ab \)
(v) \( 6a^2 - 3b^2 + 2a^2 + 5b^2 - 4a^2 \)
(vi) \( 8abc + 2ab - 4abc + ab \)
(vii) \( 9xyz + 15yxz - 10zyx - 2zxy \)
(viii) \( 13pqr + 2p + 4q - 6pqr + 5pqr \)
(ix) \( 4ab + 0 - 2ba \)
(x) \( 6x^2y - 2xy^2 + 5x^2y - xy^2 \)
(xi) \( 6.4a + 5.3b - 2.4a - 2.2b \)
(xii) \( 2.5a + 4.6b + 1.2a - 3.6b \)
(xiii) \( 22m - 12\frac{1}{2}n - 15p + 16n \)
(xiv) \( 6p + \frac{2}{3}q - 1\frac{1}{2}p + \frac{1}{3}q + 2q \)
(xv) \( 2\frac{2}{3}xy - 3\frac{1}{2}xy + 3\frac{1}{3}xy - 2\frac{1}{2}xy \)
Answer:
(i) \( 2a^2b^2 + 8a^2b^2 + 5ab^2 - 3ab^2 \)
\( \implies 10a^2b^2 + 2ab^2 \)
(ii) \( 4a - 2a + 3b - b \)
\( \implies 2a + 2b \)
(iii) \( 2xy + 5xy - 6xy + 4yz + 3yz \)
\( \implies 7xy - 6xy + 7yz \)
\( \implies xy + 7yz \)
(iv) \( ab + 15ab - 11ab - 2ab = 16ab - 13ab \)
\( \implies 3ab \)
(v) \( 6a^2 + 2a^2 - 4a^2 + 5b^2 - 3b^2 \)
\( \implies 4a^2 + 2b^2 \)
(vi) \( 8abc - 4abc + 2ab + ab \)
\( \implies 4abc + 3ab \)
(vii) \( 9xyz + 15xyz - 10xyz - 2xyz \) (since order of variables does not matter)
\( \implies 24xyz - 12xyz = 12xyz \)
(viii) \( 13pqr + 5pqr - 6pqr + 2p + 4q \)
\( \implies 18pqr - 6pqr + 2p + 4q \)
\( \implies 12pqr + 2p + 4q \)
(ix) \( 4ab - 2ab + 0 \)
\( \implies 2ab \)
(x) \( 6x^2y + 5x^2y - 2xy^2 - xy^2 \)
\( \implies 11x^2y - 3xy^2 \)
(xi) \( 6.4a - 2.4a + 5.3b - 2.2b \)
\( \implies 4a + 3.1b \)
(xii) \( 2.5a + 1.2a + 4.6b - 3.6b \)
\( \implies 3.7a + b \)
(xiii) \( 22m - \frac{25}{2}n + 16n - 15p \)
\( \implies 22m + \frac{32n - 25n}{2} - 15p \)
\( \implies 22m + \frac{7}{2}n - 15p \)
\( \implies 22m + 3\frac{1}{2}n - 15p \)
(xiv) \( 6p - \frac{3}{2}p + \frac{2}{3}q + \frac{1}{3}q + 2q \)
\( \implies \frac{12p - 3p}{2} + \left(\frac{2}{3} + \frac{1}{3} + 2\right)q \)
\( \implies \frac{9}{2}p + (1 + 2)q \)
\( \implies 4\frac{1}{2}p + 3q \)
(xv) \( xy \left( 2\frac{2}{3} - 3\frac{1}{2} + 3\frac{1}{3} - 2\frac{1}{2} \right) \)
\( \implies xy \left( \frac{8}{3} - \frac{7}{2} + \frac{10}{3} - \frac{5}{2} \right) \)
\( \implies xy \left( \frac{16 - 21 + 20 - 15}{6} \right) \)
\( \implies xy \left( \frac{36 - 36}{6} \right) = 0 \)
In simple words: When simplifying long expressions, first group terms with identical variables and exponents. Then, add or subtract their coefficients while keeping unlike terms separate.

Exam Tip: For expressions with fractions, convert any mixed numbers to improper fractions before finding a common denominator to simplify successfully.

Exercise 19(B)

Question 1. Find the sum of:
(i) \( 3a + 4b + 7c \), \( -5a + 3b - 6c \) and \( 4a - 2b - 4c \)
(ii) \( 2x^2 + xy - y^2 \), \( -x^2 + 2xy + 3y^2 \) and \( 3x^2 - 10xy + 4y^2 \)
(iii) \( x^2 - x + 1 \), \( -5x^2 + 2x - 2 \) and \( 3x^2 - 3x + 1 \)
(iv) \( a^2 - ab + bc \), \( 2ab + bc - 2a^2 \) and \( -3bc + 3a^2 + ab \)
(v) \( 4x^2 + 7 - 3x \), \( 4x - x^2 + 8 \) and \( -10 + 5x - 2x^2 \)
(vi) \( 3x + 4xy - y^2 \), \( xy - 4x + 2y^2 \) and \( 3y^2 - xy + 6x \)
Answer:
(i) \( (3a + 4b + 7c) + (-5a + 3b - 6c) + (4a - 2b - 4c) \)
\( \implies 3a + 4a - 5a + 4b + 3b - 2b + 7c - 6c - 4c \)
\( \implies 2a + 5b - 3c \)
(ii) \( (2x^2 + xy - y^2) + (-x^2 + 2xy + 3y^2) + (3x^2 - 10xy + 4y^2) \)
\( \implies 2x^2 + 3x^2 - x^2 + xy + 2xy - 10xy + 3y^2 + 4y^2 - y^2 \)
\( \implies 4x^2 - 7xy + 6y^2 \)
(iii) \( (x^2 - x + 1) + (-5x^2 + 2x - 2) + (3x^2 - 3x + 1) \)
\( \implies x^2 + 3x^2 - 5x^2 + 2x - x - 3x + 1 + 1 - 2 \)
\( \implies -x^2 - 2x \)
(iv) \( (a^2 - ab + bc) + (2ab + bc - 2a^2) + (-3bc + 3a^2 + ab) \)
\( \implies a^2 + 3a^2 - 2a^2 + 2ab + ab - ab + bc + bc - 3bc \)
\( \implies 2a^2 + 2ab - bc \)
(v) \( (4x^2 + 7 - 3x) + (4x - x^2 + 8) + (-10 + 5x - 2x^2) \)
\( \implies 4x^2 - x^2 - 2x^2 + 4x + 5x - 3x + 7 + 8 - 10 \)
\( \implies x^2 + 6x + 5 \)
(vi) \( (3x + 4xy - y^2) + (xy - 4x + 2y^2) + (3y^2 - xy + 6x) \)
\( \implies 3x - 4x + 6x + 4xy + xy - xy + 2y^2 + 3y^2 - y^2 \)
\( \implies 5x + 4xy + 4y^2 \)
In simple words: Write all the expressions in parentheses with addition signs between them. Remove the parentheses and collect matching algebraic terms together to add or subtract them.

Exam Tip: When removing parentheses, any positive sign before a set of parentheses does not change the signs inside. Grouping by descending powers of variables helps organize your work.

Question 2. Add the following expressions:
(i) \( -17x^2 - 2xy + 23y^2 \), \( -9y^2 + 15x^2 + 7xy \) and \( 13x^2 + 3y^2 - 4xy \)
(ii) \( -x^2 - 3xy + 3y^2 + 8 \), \( 3x^2 - 5y^2 - 3 + 4xy \) and \( -6xy + 2x^2 - 2 + y^2 \)
(iii) \( a^3 - 2b^3 + a \), \( b^3 - 2a^3 + b \) and \( -2b + 2b^3 - 5a + 4a^3 \)
Answer:
(i) \( (-17x^2 - 2xy + 23y^2) + (-9y^2 + 15x^2 + 7xy) + (13x^2 + 3y^2 - 4xy) \)
\( \implies -17x^2 + 15x^2 + 13x^2 - 2xy + 7xy - 4xy + 23y^2 - 9y^2 + 3y^2 \)
\( \implies 11x^2 + xy + 17y^2 \)
(ii) \( (-x^2 - 3xy + 3y^2 + 8) + (3x^2 - 5y^2 - 3 + 4xy) + (-6xy + 2x^2 - 2 + y^2) \)
\( \implies -x^2 + 3x^2 + 2x^2 - 3xy + 4xy - 6xy + 3y^2 - 5y^2 + y^2 + 8 - 3 - 2 \)
\( \implies 4x^2 - 5xy - y^2 + 3 \)
(iii) \( (a^3 - 2b^3 + a) + (b^3 - 2a^3 + b) + (-2b + 2b^3 - 5a + 4a^3) \)
\( \implies a^3 - 2a^3 + 4a^3 - 2b^3 + b^3 + 2b^3 + a - 5a + b - 2b \)
\( \implies 3a^3 + b^3 - 4a - b \)
In simple words: Add several expressions by first writing them side-by-side. Group all terms that have matching variables, and then calculate the final value of each group.

Exam Tip: Keep a neat checklist of terms when dealing with multiple variables like \( a^3 \), \( b^3 \), \( a \), and \( b \) to make sure no term is missed or duplicated.

Question 3. Evaluate:
(i) \( 3a - (a + 2b) \)
(ii) \( (5x - 3y) - (x + y) \)
(iii) \( (8a + 15b) - (3b - 7a) \)
(iv) \( (8x + 7y) - (4y - 3x) \)
(v) \( 7 - (4a - 5) \)
(vi) \( (6y - 13) - (4 - 7y) \)
Answer:
(i) \( 3a - (a + 2b) \)
When we open the brackets, the minus sign changes the sign of each term inside:
\( = 3a - a - 2b \)
Now, let's group and subtract the like terms:
\( = 2a - 2b \)
We can also write this as:
\( = 2(a - b) \)

(ii) \( (5x - 3y) - (x + y) \)
Opening the brackets gives:
\( = 5x - 3y - x - y \)
Let's bring the same variables together:
\( = 5x - x - 3y - y \)
Subtracting and simplifying:
\( = 4x - 4y \)
Factoring out the common number:
\( = 4(x - y) \)

(iii) \( (8a + 15b) - (3b - 7a) \)
Opening the brackets (remembering that minus times minus becomes plus):
\( = 8a + 15b - 3b + 7a \)
Grouping the same letters together:
\( = 8a + 7a + 15b - 3b \)
Adding and subtracting:
\( = 15a + 12b \)

(iv) \( (8x + 7y) - (4y - 3x) \)
Opening the brackets changes the signs inside the second group:
\( = 8x + 7y - 4y + 3x \)
Combining the similar terms:
\( = 8x + 3x + 7y - 4y \)
Simplifying the expression:
\( = 11x + 3y \)

(v) \( 7 - (4a - 5) \)
Removing the brackets changes the minus 5 to plus 5:
\( = 7 - 4a + 5 \)
Bringing the constant numbers together:
\( = 7 + 5 - 4a \)
Adding the numbers:
\( = 12 - 4a \)

(vi) \( (6y - 13) - (4 - 7y) \)
Opening the brackets:
\( = 6y - 13 - 4 + 7y \)
Grouping the terms:
\( = 6y + 7y - 13 - 4 \)
Simplifying:
\( = 13y - 17 \)
In simple words: When there is a minus sign outside a bracket, you must change the sign of every term inside the bracket when you open it. After that, group the same letters together and simplify.

Exam Tip: Be very careful when removing brackets with a negative sign outside. A common mistake is forgetting to change the sign of the second term inside the bracket.

Question 4. Subtract:
(i) \( 5a - 3b + 2c \) from \( a - 4b - 2c \)
(ii) \( 4x - 6y + 3z \) from \( 12x + 7y - 21z \)
(iii) \( 5 - a - 4b + 4c \) from \( 5a - 7b + 2c \)
(iv) \( -8x - 12y + 17z \) from \( x - y - z \)
(v) \( 2ab + cd - ac - 2bd \) from \( ab - 2cd + 2ac + bd \)
Answer:
(i) To subtract, we write the second expression first and then subtract the first expression from it:
\( (a - 4b - 2c) - (5a - 3b + 2c) \)
Opening the brackets and changing the signs of the terms in the second bracket:
\( = a - 4b - 2c - 5a + 3b - 2c \)
Grouping the like terms together:
\( = a - 5a - 4b + 3b - 2c - 2c \)
Simplifying the terms:
\( = -4a - b - 4c \)

(ii) We write the second expression first and subtract the first:
\( (12x + 7y - 21z) - (4x - 6y + 3z) \)
Opening the brackets changes the signs inside the second bracket:
\( = 12x + 7y - 21z - 4x + 6y - 3z \)
Grouping the same variables together:
\( = 12x - 4x + 7y + 6y - 21z - 3z \)
Simplifying:
\( = 8x + 13y - 24z \)

(iii) Subtracting the first expression from the second expression:
\( (5a - 7b + 2c) - (5 - a - 4b + 4c) \)
Removing brackets and changing signs:
\( = 5a - 7b + 2c - 5 + a + 4b - 4c \)
Grouping the identical variables:
\( = 5a + a - 7b + 4b + 2c - 4c - 5 \)
Combining the terms:
\( = 6a - 3b - 2c - 5 \)

(iv) Subtracting the first expression from the second:
\( (x - y - z) - (-8x - 12y + 17z) \)
Opening brackets (note that minus times minus becomes plus):
\( = x - y - z + 8x + 12y - 17z \)
Grouping the same variables:
\( = x + 8x + 12y - y - z - 17z \)
Combining them gives:
\( = 9x + 11y - 18z \)

(v) Subtracting the first expression from the second:
\( (ab - 2cd + 2ac + bd) - (2ab + cd - ac - 2bd) \)
Removing brackets and reversing the signs of the terms in the second bracket:
\( = ab - 2cd + 2ac + bd - 2ab - cd + ac + 2bd \)
Grouping the terms with same variables:
\( = ab - 2ab - 2cd - cd + 2ac + ac + bd + 2bd \)
Simplifying the expression:
\( = -ab - 3cd + 3ac + 3bd \)
In simple words: When we are asked to subtract "A from B", we write B first, put a minus sign, and then write A in brackets. Then, we open the brackets, change all the signs of A, and combine the like terms.

Exam Tip: Remember that "subtract A from B" always means \( B - A \), not \( A - B \). Writing the terms in the correct order is the first step to getting the correct answer.

Question 5. Take:
(i) \( -ab + bc - ca \) from \( bc - ca + ab \)
(ii) \( 5x + 6y - 3z \) from \( 3x + 5y - 4z \)
(iii) \( -\frac{3}{2}p + q - r \) from \( \frac{1}{2}p - \frac{1}{3}q - \frac{3}{2}r \)
(iv) \( 1 - a + a^2 \) from \( a^2 + a + 1 \)
Answer:
(i) "Take A from B" means to subtract A from B:
\( (bc - ca + ab) - (-ab + bc - ca) \)
Opening brackets and changing the signs:
\( = bc - ca + ab + ab - bc + ca \)
Arranging the like terms together:
\( = bc - bc - ca + ca + ab + ab \)
Simplifying, we get:
\( = 2ab \)

(ii) Subtracting the first term from the second:
\( (3x + 5y - 4z) - (5x + 6y - 3z) \)
Removing brackets and changing signs:
\( = 3x + 5y - 4z - 5x - 6y + 3z \)
Grouping like variables:
\( = 3x - 5x + 5y - 6y - 4z + 3z \)
Simplifying:
\( = -2x - y - z \)

(iii) Subtracting the first expression from the second:
\( \left(\frac{1}{2}p - \frac{1}{3}q - \frac{3}{2}r\right) - \left(-\frac{3}{2}p + q - r\right) \)
Opening the brackets:
\( = \frac{1}{2}p - \frac{1}{3}q - \frac{3}{2}r + \frac{3}{2}p - q + r \)
Grouping the same variables:
\( = \frac{1}{2}p + \frac{3}{2}p - \frac{1}{3}q - q - \frac{3}{2}r + r \)
Taking the Lowest Common Multiple (L.C.M.) as 6 to combine the fractions:
\( = \frac{3p + 9p - 2q - 6q - 9r + 6r}{6} \)
\( = \frac{12p}{6} - \frac{8q}{6} - \frac{3r}{6} \)
Reducing the fractions to their simplest forms:
\( = 2p - \frac{4}{3}q - \frac{1}{2}r \)

(iv) Subtracting the first expression from the second:
\( (a^2 + a + 1) - (1 - a + a^2) \)
Removing the brackets:
\( = a^2 + a + 1 - 1 + a - a^2 \)
Grouping the like powers of \( a \) and constants:
\( = a^2 - a^2 + a + a + 1 - 1 \)
Simplifying the terms:
\( = 2a \)
In simple words: The term "Take A from B" simply means to perform subtraction: \( B - A \). Open the brackets carefully, especially when there is a negative sign, and combine like terms. For fractional terms, find a common denominator first.

Exam Tip: For expressions involving fractions, finding the L.C.M. of all denominators is a very clean way to perform the addition or subtraction without making arithmetic errors.

Question 6. From the sum of \( x + y - 2z \) and \( 2x - y + z \) subtract \( x + y + z \).
Answer:
First, we add the first two expressions and then subtract the third one from their sum:
\( (x + y - 2z) + (2x - y + z) - (x + y + z) \)
Removing the brackets and applying the signs:
\( = x + y - 2z + 2x - y + z - x - y - z \)
Now, let's group the terms of \( x \), \( y \), and \( z \) together:
\( = x + 2x - x + y - y - y - 2z + z - z \)
Simplifying each group:
\( = 2x - y - 2z \)
In simple words: First, combine the first two expressions by adding them. Then, subtract the third expression from that total. Remember to change the sign of each term in the expression you are subtracting.

Exam Tip: Break the problem into two logical steps - first perform the addition, then perform the subtraction. This prevents sign confusion and keeps the calculation clean.

Question 7. From the sum of \( 3a - 2b + 4c \) and \( 3b - 2c \) subtract \( a - b - c \).
Answer:
We begin by adding the first two expressions and then subtracting the third expression:
\( (3a - 2b + 4c) + (3b - 2c) - (a - b - c) \)
Opening all the brackets (changing the signs of the terms in the subtracted bracket):
\( = 3a - 2b + 4c + 3b - 2c - a + b + c \)
Grouping the terms by their variables:
\( = 3a - a + 3b + b - 2b + 4c + c - 2c \)
Simplifying the grouped terms:
\( = 2a + 2b + 3c \)
In simple words: Add the first two algebraic expressions together. After that, subtract the last expression from the sum by reversing its signs and combining similar terms.

Exam Tip: Be very careful when grouping terms. Tick or cross out each term as you write it down to ensure no term is missed or duplicated in the next step.

Question 8. Subtract \( x - 2y - z \) from the sum of \( 3x - y + z \) and \( x + y - 3z \).
Answer:
First, we find the sum of the last two expressions and then subtract the first expression from it:
\( (3x - y + z) + (x + y - 3z) - (x - 2y - z) \)
Opening the brackets and adjusting the signs:
\( = 3x - y + z + x + y - 3z - x + 2y + z \)
Gathering the same variables together:
\( = 3x + x - x - y + y + 2y + z + z - 3z \)
Simplifying the terms:
\( = 3x + 2y - z \)
In simple words: Add the second and third expressions. Then, subtract the first expression from this sum by changing its signs and combining all like terms.

Exam Tip: Always read the phrasing of the question carefully to identify which expressions are being added and which one is being subtracted.

Question 9. Subtract the sum of \( x + y \) and \( x - z \) from the sum of \( x - 2z \) and \( x + y + z \).
Answer:
We write the subtraction of the first sum from the second sum:
\( \left[(x - 2z) + (x + y + z)\right] - \left[(x + y) + (x - z)\right] \)
Opening the inner brackets first:
\( = x - 2z + x + y + z - (x + y + x - z) \)
Now, opening the outer brackets by changing the signs of the terms inside the subtracted part:
\( = x - 2z + x + y + z - x - y - x + z \)
Grouping the similar variables:
\( = x + x - x - x + y - y + z + z - 2z \)
Simplifying the grouped terms:
\( = 0 \)
In simple words: Calculate the sum of the first two expressions, and the sum of the next two expressions. Then, subtract the first sum from the second sum. In this case, all the terms cancel out, leaving us with zero.

Exam Tip: When dealing with multiple sums and subtractions, use curly brackets or square brackets to keep the two parts separate before performing the final subtraction.

Question 10. By how much should \( x + 2y - 3z \) be increased to get \( 3x \)?
Answer:
To find out how much the given expression needs to be increased, we subtract it from the target value:
\( 3x - (x + 2y - 3z) \)
Removing the brackets and changing the signs of each term inside:
\( = 3x - x - 2y + 3z \)
Subtracting the terms of \( x \):
\( = 2x - 2y + 3z \)
In simple words: If you want to know what to add to a number to reach a target, subtract that number from the target. Here, we subtract the starting expression from \( 3x \).

Exam Tip: Treat algebraic "increased to get" questions like basic arithmetic. For example, "by how much should 3 be increased to get 10" is \( 10 - 3 = 7 \). This helps you set up the correct subtraction order.

Question 11. The sum of two expressions is \( 5x^2 - 3y^2 \). If one of them is \( 3x^2 + 4xy - y^2 \), find the other.
Answer:
To find the other expression, we subtract the given expression from the total sum:
\( (5x^2 - 3y^2) - (3x^2 + 4xy - y^2) \)
Removing brackets and changing signs of the terms being subtracted:
\( = 5x^2 - 3y^2 - 3x^2 - 4xy + y^2 \)
Grouping the like terms together:
\( = 5x^2 - 3x^2 - 4xy - 3y^2 + y^2 \)
Simplifying the terms:
\( = 2x^2 - 4xy - 2y^2 \)
In simple words: If you know the sum of two numbers and one of the numbers, you subtract that number from the sum to find the missing one.

Exam Tip: Be careful with the signs when subtracting the second term of the expression (like \( -y^2 \) which becomes \( +y^2 \) after subtraction). This is a common place where signs are miscalculated.

Question 12. The sum of two expressions is \( 3a^2 + 2ab - b^2 \). If one of them is \( 2a^2 + 3b^2 \), find the other.
Answer:
We subtract the known expression from the total sum to find the other expression:
\( (3a^2 + 2ab - b^2) - (2a^2 + 3b^2) \)
Opening the brackets and reversing the signs inside the subtracted expression:
\( = 3a^2 + 2ab - b^2 - 2a^2 - 3b^2 \)
Grouping similar terms:
\( = 3a^2 - 2a^2 + 2ab - b^2 - 3b^2 \)
Combining like terms:
\( = a^2 + 2ab - 4b^2 \)
In simple words: Subtract the given expression from the sum to find the missing expression. Group the similar terms to get the final simplified answer.

Exam Tip: Since there is no term with \( ab \) in the second expression, it remains unchanged during subtraction. Do not try to combine it with terms containing different variables like \( a^2 \) or \( b^2 \).

Exercise 19(C)

Question 1. Fill in the blanks:
(i) \( 6 \times 3 = \) ........ and \( 6x \times 3x = \) ...........
(ii) \( 6 \times 3 = \) ........ and \( 6x^2 \times 3x^3 = \) ...........
(iii) \( 5 \times 4 = \) ........ and \( 5x \times 4y = \) ...........
(iv) \( 4 \times 7 = \) ........ and \( 4ax \times 7x = \) ...........
(v) \( 6 \times 2 = \) ........ and \( 6xy \times 2xy = \) ...........
(vi) \( 12 \times 4 = \) ........ and \( 12ax^2 \times 4ax = \) ...........
(vii) \( 1 \times 8 = \) ........ and \( a^2xy^2 \times 8a^3x^2y = \) ...........
(viii) \( 15 \times 3 = \) ........ and \( 15x \times 3x^5y^2 = \) ...........
Answer:
(i) \( 6 \times 3 = 18 \) and \( 6x \times 3x = 18x^2 \)
(ii) \( 6 \times 3 = 18 \) and \( 6x^2 \times 3x^3 = 18x^5 \)
(iii) \( 5 \times 4 = 20 \) and \( 5x \times 4y = 20xy \)
(iv) \( 4 \times 7 = 28 \) and \( 4ax \times 7x = 28ax^2 \)
(v) \( 6 \times 2 = 12 \) and \( 6xy \times 2xy = 12x^2y^2 \)
(vi) \( 12 \times 4 = 48 \) and \( 12ax^2 \times 4ax = 48a^2x^3 \)
(vii) \( 1 \times 8 = 8 \) and \( a^2xy^2 \times 8a^3x^2y = 8a^5x^3y^3 \)
(viii) \( 15 \times 3 = 45 \) and \( 15x \times 3x^5y^2 = 45x^6y^2 \)
In simple words: When multiplying terms with variables, multiply the numbers first. Then, add the exponents of the same variables to find the final power.

Exam Tip: Remember the rule of exponents: \( x^m \times x^n = x^{m+n} \). When a variable has no written power, its exponent is 1 (e.g., \( x = x^1 \)).

Question 2. Fill in the blanks:
(i) \( 4x \times 6x \times 2 = \) ...........
(ii) \( 3ab \times 6ax = \) ...........
(iii) \( x \times 2x^2 \times 3x^3 = \) ...........
(iv) \( 5 \times 5a^3 = \) ...........
(v) \( 6 \times 6x^2 \times 6x^2y^2 = \) ...........
(vi) \( -8x \times -3x = \) ...........
(vii) \( -5 \times -3x \times 5x^2 = \) ...........
(viii) \( 8 \times -4xy^2 \times 3x^3y^2 = \) ...........
(ix) \( -4x \times 5xy \times 3z = \) ...........
(x) \( 5x \times 2x^2y \times (-7y^3) \times 2x^3y^2 = \) ...........
Answer:
(i) \( 4x \times 6x \times 2 = 4 \times 6 \times 2 \times x \times x = 48x^2 \)
(ii) \( 3ab \times 6ax = 3 \times 6 \times a \times a \times b \times x = 18a^2bx \)
(iii) \( x \times 2x^2 \times 3x^3 = 1 \times 2 \times 3 \times x^{1+2+3} = 6x^6 \)
(iv) \( 5 \times 5a^3 = 25a^3 \)
(v) \( 6 \times 6x^2 \times 6x^2y^2 = 6 \times 6 \times 6 \times x^{2+2}y^2 = 216x^4y^2 \)
(vi) \( -8x \times -3x = (-8 \times -3) \times x^{1+1} = 24x^2 \)
(vii) \( -5 \times -3x \times 5x^2 = (-5 \times -3 \times 5) \times x^{1+2} = 75x^3 \)
(viii) \( 8 \times -4xy^2 \times 3x^3y^2 = (8 \times -4 \times 3) \times x^{1+3}y^{2+2} = -96x^4y^4 \)
(ix) \( -4x \times 5xy \times 3z = (-4 \times 5 \times 3) \times x^{1+1}yz = -60x^2yz \)
(x) \( 5x \times 2x^2y \times (-7y^3) \times 2x^3y^2 = (5 \times 2 \times -7 \times 2) \times x^{1+2+3}y^{1+3+2} = -140x^6y^6 \)
In simple words: To multiply algebraic terms, first multiply all the numerical coefficients together. Then, multiply the variables by adding up the powers of identical letters.

Exam Tip: Be very careful with positive and negative signs. Remember that multiplying two negative numbers gives a positive result, while multiplying a positive and a negative number gives a negative result.

Question 3. Find the value of:
(i) \( 3x^3 \times 5x^4 \)
(ii) \( 5a^2 \times 7a^7 \)
(iii) \( 3abc \times 6ac^3 \)
(iv) \( a^2b^2 \times 5a^3b^4 \)
(v) \( 2x^2y^3 \times 5x^3y^4 \)
(vi) \( abc \times bcd \)
Answer:
(i) \( 3x^3 \times 5x^4 = 3 \times 5 \times x^{3+4} = 15x^7 \)
(ii) \( 5a^2 \times 7a^7 = 5 \times 7 \times a^{2+7} = 35a^9 \)
(iii) \( 3abc \times 6ac^3 = (3 \times 6) \times a^{1+1} \times b \times c^{1+3} = 18a^2bc^4 \)
(iv) \( a^2b^2 \times 5a^3b^4 = 5 \times a^{2+3} \times b^{2+4} = 5a^5b^6 \)
(v) \( 2x^2y^3 \times 5x^3y^4 = (2 \times 5) \times x^{2+3} \times y^{3+4} = 10x^5y^7 \)
(vi) \( abc \times bcd = a \times b^{1+1} \times c^{1+1} \times d = ab^2c^2d \)
In simple words: To multiply variables, write down the common variable and add up their powers. For any standard numbers, multiply them directly.

Exam Tip: Don't forget that a variable with no exponent shown (like \( b \) or \( c \)) has an implicit power of 1. Always write it as \( b^1 \) in your head so you don't forget to add 1 to the exponent of the other terms.

Question 4. Multiply:
(i) \( a + b \) by \( ab \)
(ii) \( 3ab - 4b \) by \( 3ab \)
(iii) \( 2xy - 5by \) by \( 4bx \)
(iv) \( 4x + 2y \) by \( 3xy \)
(v) \( x^2 - x \) by \( 2x \)
(vi) \( 1 + 4x \) by \( x \)
(vii) \( 9xy^2 + 3x^2y \) by \( 5xy \)
(viii) \( 6x - 5y \) by \( 3axy \)
Answer:
(i) \( (a + b) \times ab = a(ab) + b(ab) = a^2b + ab^2 \)
(ii) \( (3ab - 4b) \times 3ab = 3ab(3ab) - 4b(3ab) = 9a^2b^2 - 12ab^2 \)
(iii) \( (2xy - 5by) \times 4bx = 2xy(4bx) - 5by(4bx) = 8bx^2y - 20b^2xy \)
(iv) \( (4x + 2y) \times 3xy = 4x(3xy) + 2y(3xy) = 12x^2y + 6xy^2 \)
(v) \( (x^2 - x) \times 2x = x^2(2x) - x(2x) = 2x^3 - 2x^2 \)
(vi) \( (1 + 4x) \times x = 1(x) + 4x(x) = x + 4x^2 \)
(vii) \( (9xy^2 + 3x^2y) \times 5xy = 9xy^2(5xy) + 3x^2y(5xy) = 45x^2y^3 + 15x^3y^2 \)
(viii) \( (6x - 5y) \times 3axy = 6x(3axy) - 5y(3axy) = 18ax^2y - 15axy^2 \)
In simple words: To multiply a binomial (an expression with two terms) by a monomial (one term), distribute the monomial to each of the two terms inside the expression.

Exam Tip: Remember the distributive property: \( m(a + b) = ma + mb \). Ensure that you multiply the term outside the parentheses with both terms inside, not just the first one.

Question 5. Multiply:
(i) \( -x + y - z \) and \( -2x \)
(ii) \( xy - yz \) and \( x^2yz^2 \)
(iii) \( 2xyz + 3xy \) and \( -2y^2z \)
(iv) \( -3xy^2 + 4x^2y \) and \( -xy \)
(v) \( 4xy \) and \( -x^2y - 3x^2y^2 \)
Answer:
(i) \( (-x + y - z) \times (-2x) = -x(-2x) + y(-2x) - z(-2x) = 2x^2 - 2xy + 2xz \)
(ii) \( (xy - yz) \times x^2yz^2 = xy(x^2yz^2) - yz(x^2yz^2) = x^3y^2z^2 - x^2y^2z^3 \)
(iii) \( (2xyz + 3xy) \times (-2y^2z) = 2xyz(-2y^2z) + 3xy(-2y^2z) = -4xy^3z^2 - 6xy^3z \)
(iv) \( (-3xy^2 + 4x^2y) \times (-xy) = -3xy^2(-xy) + 4x^2y(-xy) = 3x^2y^3 - 4x^3y^2 \)
(v) \( (-x^2y - 3x^2y^2) \times 4xy = -x^2y(4xy) - 3x^2y^2(4xy) = -4x^3y^2 - 12x^3y^3 \)
In simple words: Multiply each term inside the bracket by the monomial outside. Be very careful with signs: negative times negative equals positive, and positive times negative equals negative.

Exam Tip: When multiplying complex variables, keep track of each letter separately. For example, for \( xy \times x^2yz^2 \), handle \( x \), \( y \), and \( z \) one by one to ensure you don't miss any exponent updates.

Question 6. Multiply:
(i) \( 3a + 4b - 5c \) and \( 3a \)
(ii) \( -5xy \) and \( -xy^2 - 6x^2y \)
Answer:
(i) \( (3a + 4b - 5c) \times 3a = (3a \times 3a) + (4b \times 3a) - (5c \times 3a) = 9a^2 + 12ab - 15ac \)
(ii) \( (-xy^2 - 6x^2y) \times -5xy = -xy^2(-5xy) - 6x^2y(-5xy) = 5x^2y^3 + 30x^3y^2 \)
In simple words: Distribute the multiplying term outside to each of the terms inside the parentheses. Sum or subtract the resulting products.

Exam Tip: Always double check the signs when multiplying a negative outside term with negative terms inside the brackets. Negative times negative gives positive.

Question 7. Multiply :
(i) x + 2 and x + 10
(ii) x + 5 and x - 3
(iii) x - 5 and x + 3
(iv) x - 5 and x - 3
(v) 2x + y and x + 3y
(vi) (3x - 5y) and (x + 6y)
(vii) (x + 9y) and (x - 5y)
(viii) (2x + 5y) and (2x + 5y)
Answer:
(i) \( (x + 2)(x + 10) \)
\( = x(x + 2) + 10(x + 2) \)
\( = x^2 + 2x + 10x + 20 \)
\( = x^2 + 12x + 20 \)

(ii) \( (x + 5)(x - 3) \)
\( = x(x + 5) - 3(x + 5) \)
\( = x^2 + 5x - 3x - 15 \)
\( = x^2 + 2x - 15 \)

(iii) \( (x - 5)(x + 3) \)
\( = x(x - 5) + 3(x - 5) \)
\( = x^2 - 5x + 3x - 15 \)
\( = x^2 - 2x - 15 \)

(iv) \( (x - 5)(x - 3) \)
\( = x(x - 5) - 3(x - 5) \)
\( = x^2 - 5x - 3x + 15 \)
\( = x^2 - 8x + 15 \)

(v) \( (2x + y)(x + 3y) \)
\( = x(2x + y) + 3y(2x + y) \)
\( = 2x^2 + xy + 6xy + 3y^2 \)
\( = 2x^2 + 7xy + 3y^2 \)

(vi) \( (3x - 5y)(x + 6y) \)
\( = x(3x - 5y) + 6y(3x - 5y) \)
\( = 3x^2 - 5xy + 18xy - 30y^2 \)
\( = 3x^2 + 13xy - 30y^2 \)

(vii) \( (x + 9y)(x - 5y) \)
\( = x(x + 9y) - 5y(x + 9y) \)
\( = x^2 + 9xy - 5xy - 45y^2 \)
\( = x^2 + 4xy - 45y^2 \)

(viii) \( (2x + 5y)(2x + 5y) \)
\( = 2x(2x + 5y) + 5y(2x + 5y) \)
\( = 4x^2 + 10xy + 10xy + 25y^2 \)
\( = 4x^2 + 20xy + 25y^2 \)
In simple words: To multiply two algebraic expressions, multiply each term in the first bracket by each term in the second bracket. Then, group the like terms together and simplify them.

Exam Tip: Pay close attention to negative signs during multiplication, as multiplying two negative terms always results in a positive term.

Question 8. Multiply :
(i) \( 3abc \) and \( -5a^2b^2c \)
(ii) \( x - y + z \) and \( -2x \)
(iii) \( 2x - 3y - 5z \) and \( -2y \)
(iv) \( -8xyz + 10x^2yz^3 \) and \( xyz \)
(v) \( xyz \) and \( -13xy^2z + 15x^2yz - 6xyz^2 \)
(vi) \( 4abc - 5a^2bc - 6ab^2c \) and \( -2abc^2 \)
Answer:
(i) \( 3abc \times (-5a^2b^2c) \)
\( = 3 \times (-5) \cdot a^{1+2}b^{1+2}c^{1+1} \)
\( = -15a^3b^3c^2 \)

(ii) \( (x - y + z) \cdot (-2x) \)
\( = -2x^2 + 2xy - 2xz \)

(iii) \( (2x - 3y - 5z) \cdot (-2y) \)
\( = -4xy + 6y^2 + 10yz \)

(iv) \( (-8xyz + 10x^2yz^3) \cdot xyz \)
\( = -8x^2y^2z^2 + 10x^3y^2z^4 \)

(v) \( (-13xy^2z + 15x^2yz - 6xyz^2) \cdot xyz \)
\( = -13x^2y^3z^2 + 15x^3y^2z^2 - 6x^2y^2z^3 \)

(vi) \( (4abc - 5a^2bc - 6ab^2c) \cdot (-2abc^2) \)
\( = -8a^2b^2c^3 + 10a^3b^2c^3 + 12a^2b^3c^3 \)
In simple words: When multiplying a term with several variables by another term, multiply their numbers first. Then, add the exponents of the same variables together.

Exam Tip: Remember that any variable without a written exponent has an implicit power of 1, which must be added when multiplying.

Question 9. Find the product of :
(i) \( xy - ab \) and \( xy + ab \)
(ii) \( 2abc - 3xy \) and \( 2abc + 3xy \)
(iii) \( a + b - c \) and \( 2a - 3b \)
(iv) \( 5x - 6y - 7z \) and \( 2x + 3y \)
(v) \( 5x - 6y - 7z \) and \( 2x + 3y + z \)
(vi) \( 2a + 3b - 4c \) and \( a - b - c \)
Answer:
(i) \( (xy - ab)(xy + ab) \)
\( = xy(xy - ab) + ab(xy - ab) \)
\( = x^2y^2 - abxy + abxy - a^2b^2 \)
\( = x^2y^2 - a^2b^2 \)

(ii) \( (2abc - 3xy)(2abc + 3xy) \)
\( = 2abc(2abc - 3xy) + 3xy(2abc - 3xy) \)
\( = 4a^2b^2c^2 - 6abcxy + 6abcxy - 9x^2y^2 \)
\( = 4a^2b^2c^2 - 9x^2y^2 \)

(iii) \( (a + b - c)(2a - 3b) \)
\( = 2a(a + b - c) - 3b(a + b - c) \)
\( = 2a^2 + 2ab - 2ac - 3ab - 3b^2 + 3bc \)
\( = 2a^2 - ab - 2ac - 3b^2 + 3bc \)
\( = 2a^2 - ab - 2ac + 3bc - 3b^2 \)

(iv) \( (5x - 6y - 7z)(2x + 3y) \)
\( = 2x(5x - 6y - 7z) + 3y(5x - 6y - 7z) \)
\( = 10x^2 - 12xy - 14xz + 15xy - 18y^2 - 21yz \)
\( = 10x^2 + 3xy - 14xz - 18y^2 - 21yz \)

(v) \( (5x - 6y - 7z)(2x + 3y + z) \)
\( = 2x(5x - 6y - 7z) + 3y(5x - 6y - 7z) + z(5x - 6y - 7z) \)
\( = 10x^2 - 12xy - 14xz + 15xy - 18y^2 - 21yz + 5xz - 6yz - 7z^2 \)
\( = 10x^2 - 12xy + 15xy - 14xz + 5xz - 18y^2 - 21yz - 6yz - 7z^2 \)
\( = 10x^2 + 3xy - 9xz - 18y^2 - 27yz - 7z^2 \)

(vi) \( (2a + 3b - 4c)(a - b - c) \)
\( = a(2a + 3b - 4c) - b(2a + 3b - 4c) - c(2a + 3b - 4c) \)
\( = 2a^2 + 3ab - 4ac - 2ab - 3b^2 + 4bc - 2ac - 3bc + 4c^2 \)
\( = 2a^2 + 3ab - 2ab - 4ac - 2ac - 3b^2 + 4bc - 3bc + 4c^2 \)
\( = 2a^2 + ab - 6ac - 3b^2 + bc + 4c^2 \)
In simple words: To multiply a longer expression by another bracket, multiply every single part of the first expression by each part of the second. Finally, combine terms that have the exact same variables.

Exam Tip: Be methodical and write out each step when expanding brackets with three or more terms to avoid leaving out any cross-products.

Exercise 19(D)

Question 1. Divide :
(i) 3a by a
(ii) 15x by 3x
(iii) 16m by 4
(iv) 20x² by 5x
(v) 30p² by 10p²
(vi) 14a³b³ by 2a²
(vii) 18pqr² by 3pq
(viii) 100 by 50b
Answer:
(i) \( 3a \div a = \frac{3 \times a}{a} = 3 \)

(ii) \( 15x \div 3x = \frac{3 \times 5 \times x}{3 \times x} = 5 \)

(iii) \( 16m \div 4 = \frac{4 \times 4 \times m}{4} = 4m \)

(iv) \( 20x^2 \div 5x = \frac{4 \times 5 \times x^{2-1}}{5} = 4x \)

(v) \( 30p^2 \div 10p^2 = \frac{3 \times 10p^2}{10p^2} = 3 \)

(vi) \( 14a^3b^3 \div 2a^2 = \frac{2 \times 7a^{3-2}b^3}{2} = 7ab^3 \)

(vii) \( 18pqr^2 \div 3pq = \frac{3 \times 6 \times p \times q \times r^2}{3 \times p \times q} = 6r^2 \)

(viii) \( 100 \div 50b = \frac{2 \times 50}{50 \times b} = \frac{2}{b} \)
In simple words: To divide terms, divide their coefficients and subtract the powers of matching variables in the denominator from those in the numerator.

Exam Tip: When dividing identical variables, subtract the lower exponent from the upper exponent using the rules of indices.

Question 2. Simplify :
(i) 2x⁵ ÷ x²
(ii) 6a⁸ ÷ 3a³
(iii) 20xy ÷ -5xy
(iv) -24a²b²c² ÷ 6ab
(v) -5x²y ÷ xy²
(vi) 40p³q⁴r⁵ ÷ 10p³q
(vii) -64x⁴y³z ÷ 4x³y²z
(viii) 35xy⁵ ÷ 7x²y⁴
Answer:
(i) \( 2x^5 \div x^2 = \frac{2x^5}{x^2} = 2x^{5-2} = 2x^3 \)

(ii) \( 6a^8 \div 3a^3 = \frac{2 \times 3 \times a^{8-3}}{3} = 2a^5 \)

(iii) \( 20xy \div (-5xy) = \frac{4 \times 5 \times x \times y}{-5 \times x \times y} = -4 \)

(iv) \( -24a^2b^2c^2 \div 6ab = \frac{-4 \times 6 \times a^{2-1}b^{2-1}c^2}{6} = -4abc^2 \)

(v) \( -5x^2y \div xy^2 = \frac{-5x^{2-1}}{y^{2-1}} = -\frac{5x}{y} \)

(vi) \( 40p^3q^4r^5 \div 10p^3q = \frac{4 \times 10 \times p^{3-3} \cdot q^{4-1} \cdot r^5}{10} = 4q^3r^5 \)

(vii) \( -64x^4y^3z \div 4x^3y^2z = \frac{-4 \times 4 \times 4 \times x^4 \times y^3 \times z}{4 \times x^3 \times y^2 \times z} = -16x^{4-3}y^{3-2} = -16xy \)

(viii) \( 35xy^5 \div 7x^2y^4 = \frac{5 \times 7 \times y^{5-4}}{7 \times x^{2-1}} = \frac{5y}{x} \)
In simple words: When simplifying division of algebraic terms, reduce the numerical coefficients and cancel out or subtract exponents of common variables.

Exam Tip: Be careful with signs; dividing a negative term by a positive term results in a negative quotient, while dividing two negatives gives a positive result.

Question 3. Divide :
(i) \( -\frac{3m}{4} \) by \( 2m \)
(ii) \( -15p^6q^8 \) by \( -5p^6q^7 \)
(iii) \( -21m^5n^7 \) by \( 14m^2n^2 \)
(iv) \( 36a^4x^5y^6 \) by \( 4x^2a^3y^2 \)
(v) \( 20x^3a^6 \) by \( 5xy \)
(vi) \( \frac{28a^2b^3}{c^2} \) by \( 4abc \)
(vii) \( \frac{2a^2}{9b^2} \) by \( \frac{3b}{2a} \)
(viii) \( \frac{-5.5x^2}{y} \) by \( \frac{11x}{y} \)
(ix) \( \frac{64x^2y^2}{z^2} \) by \( \frac{8xy}{z} \)
Answer:
(i) \( -\frac{3m}{4} \div 2m = \frac{-3 \times m}{4 \times 2 \times m} = -\frac{3}{8} \)

(ii) \( -15p^6q^8 \div (-5p^6q^7) = \frac{-5 \times 3 \times p^6 \times q^8}{-5 \times p^6 \times q^7} = 3q^{8-7} = 3q \)

(iii) \( -21m^5n^7 \div 14m^2n^2 = \frac{-3 \times 7 \times m^{5-2} \cdot n^{7-2}}{14} = -\frac{3}{2}m^3n^5 \)

(iv) \( 36a^4x^5y^6 \div 4x^2a^3y^2 = \frac{4 \times 9 \times a^{4-3} \cdot x^{5-2} \cdot y^{6-2}}{4} = 9ax^3y^4 \)

(v) \( 20x^3a^6 \div 5xy = \frac{4 \times 5 \times x^{3-1} \cdot a^6}{5xy} = \frac{4x^2a^6}{y} \)

(vi) \( \frac{28a^2b^3}{c^2} \div 4abc = \frac{4 \times 7 \times a^{2-1} \cdot b^{3-1}}{4 \times c^{2+1}} = \frac{7ab^2}{c^3} \)

(vii) \( \frac{2a^2}{9b^2} \div \frac{3b}{2a} = \frac{2a^2}{9b^2} \times \frac{2a}{3b} = \frac{2 \times 2 \times a^{2+1}}{9 \times 3 \cdot b^{2+1}} = \frac{4a^3}{27b^3} \)

(viii) \( \frac{-5.5x^2}{y} \div \frac{11x}{y} = \frac{-55x^2}{10y} \times \frac{y}{11x} = -\frac{5x}{10} = -0.5x \)

(ix) \( \frac{64x^2y^2}{z^2} \div \frac{8xy}{z} = \frac{8 \times 8 \times x^2 \times y^2}{z^2} \times \frac{z}{8 \times x \times y} = \frac{8x^{2-1}y^{2-1}}{z^{2-1}} = \frac{8xy}{z} \)
In simple words: To divide fractions in algebra, multiply the first fraction by the reciprocal of the second fraction, then cancel out any common factors.

Exam Tip: Always remember to invert the divisor (the second term or fraction) when performing division with algebraic fractions.

Question 4. Simplify :
(i) \( \frac{-15m^5n^2}{-3m^5} \)
(ii) \( \frac{35x^4y^2}{-15x^2y^2} \)
(iii) \( \frac{-24x^6y^2}{6x^6y} \)
Answer:
(i) \( \frac{-15m^5n^2}{-3m^5} = \frac{-3 \times 5 \times m^5 \times n^2}{-3 \times m^5} = 5n^2 \)

(ii) \( \frac{35x^4y^2}{-15x^2y^2} = \frac{-5 \times -7 \times x^4 \times y^2}{3 \times -5 \times x^2 \times y^2} = -\frac{7x^{4-2}}{3} = -\frac{7x^2}{3} \)

(iii) \( \frac{-24x^6y^2}{6x^6y} = \frac{-4 \times 6 \times x^6 \times y^2}{6 \times x^6 \times y} = -4y^{2-1} = -4y \)
In simple words: Simplify these fractions by dividing the numbers and cancelling common variables in the top and bottom.

Exam Tip: If variables in both the numerator and denominator have the exact same power, they cancel out completely and leave a factor of 1.

Question 5. Divide :
(i) 9x³ - 6x² by 3x
(ii) 6m² - 16m³ + 10m⁴ by -2m
(iii) 15x³y² + 25x²y³ - 36x⁴y⁴ by 5x²y²
(iv) 36a³x⁵ - 24a⁴x⁴ + 18a⁵x³ by -6a³x³
Answer:
(i) \( \frac{9x^3 - 6x^2}{3x} = \frac{9x^3}{3x} - \frac{6x^2}{3x} \)
\( = 3x^{3-1} - 2x^{2-1} \)
\( = 3x^2 - 2x \)

(ii) \( \frac{6m^2 - 16m^3 + 10m^4}{-2m} = \frac{6m^2}{-2m} - \frac{16m^3}{-2m} + \frac{10m^4}{-2m} \)
\( = -3m^{2-1} - (-8m^{3-1}) - 5m^{4-1} \)
\( = -3m + 8m^2 - 5m^3 \)

(iii) \( \frac{15x^3y^2 + 25x^2y^3 - 36x^4y^4}{5x^2y^2} = \frac{15x^3y^2}{5x^2y^2} + \frac{25x^2y^3}{5x^2y^2} - \frac{36x^4y^4}{5x^2y^2} \)
\( = 3x^{3-2}y^{2-2} + 5x^{2-2}y^{3-2} - \frac{36}{5}x^{4-2}y^{4-2} \)
\( = 3x^1y^0 + 5x^0y^1 - \frac{36}{5}x^2y^2 \)
\( = 3x + 5y - \frac{36}{5}x^2y^2 \) (Since \( x^0 = 1 \) and \( y^0 = 1 \))

(iv) \( \frac{36a^3x^5 - 24a^4x^4 + 18a^5x^3}{-6a^3x^3} = \frac{36a^3x^5}{-6a^3x^3} - \frac{24a^4x^4}{-6a^3x^3} + \frac{18a^5x^3}{-6a^3x^3} \)
\( = -6a^{3-3}x^{5-3} - (-4a^{4-3}x^{4-3}) + (-3a^{5-3}x^{3-3}) \)
\( = -6a^0x^2 + 4a^1x^1 - 3a^2x^0 \)
\( = -6x^2 + 4ax - 3a^2 \) (Since \( a^0 = 1 \) and \( x^0 = 1 \))
In simple words: When dividing a polynomial by a single term (monomial), divide every individual term in the polynomial by that term and simplify each part.

Exam Tip: Remember that any variable raised to the power of 0 becomes equal to 1, which helps in simplifying the final expression.