RS Aggarwal Class 8 Mathematics Solutions Chapter 6 Operations on Algebraic Expressions

Exercise 6A

Exam Tip: When adding or subtracting algebraic expressions, always arrange like terms in rows and combine them column by column to avoid errors.

Question 1. Add the following expressions:
(i) \( 8ab, -5ab, 3ab, -ab \)
(ii) \( 7x, -3x, 5x, -x, -2x \)
(iii) \( 3a - 4b + 4c, 2a + 3b - 8c, a - 6b + c \)
Answer:
(i) Writing the given expressions in the same order as rows with matching terms below each other and combining column by column yields \( 5ab \)
(ii) Writing the given expressions in the same order as rows with matching terms below each other and combining column by column yields \( 6x \)
(iii) Writing the given expressions in the same order as rows with matching terms below each other and combining column by column yields \( 6a - 7b - 3c \)

Exam Tip: Combine coefficients of the same variable carefully; keep track of positive and negative signs throughout.

Question 2. Add the following expressions:
(i) \( 5x - 8y + 2z, -2x - 4y + 3z, -x + 6y - z \)
(ii) \( 6ax - 2by + 3cz, -11ax + 6by - cz, -2ax - 3by + 10cz \)
Answer:
(i) Writing the given expressions in the same order as rows with matching terms below each other and combining column by column yields \( 5x - 9y + 2z \)
(ii) Writing the given expressions in the same order as rows with matching terms below each other and combining column by column yields \( -7ax + by + 12cz \)

Exam Tip: Group terms by their variables first; this prevents sign errors when multiple terms have the same variable.

Question 3. Add the following expressions:
(i) \( 3a - 4b + 4c, 2a + 3b - 8c, a - 6b + c \)
(ii) \( 2a + 3b - 8c, a - 6b + c, 3a - 4b + 4c \)
Answer:
(i) Writing the given expressions in the same order as rows with matching terms below each other and combining column by column yields \( 6a - 7b - 3c \)
(ii) The order of the three expressions is different, but the final result remains the same: \( 6a - 7b - 3c \)

Exam Tip: The commutative property of addition means the order in which you add expressions does not change the final answer.

Question 4. Add the following expressions:
(i) \( 5x - 8y + 2z, -2x - 4y + 3z, -x + 6y - z \)
(ii) \( 6ax - 2by + 3cz, -11ax + 6by - cz, -2ax - 3by + 10cz \)
Answer:
(i) Writing the given expressions in the same order as rows with matching terms below each other and combining column by column yields \( 5x - 9y + 2z \)
(ii) Writing the given expressions in the same order as rows with matching terms below each other and combining column by column yields \( -7ax + by + 12cz \)

Exam Tip: Always verify your final answer by selecting one variable and checking that all its terms have been combined correctly.

Question 5. Simplify by adding:
(i) \( 6ax - 2by + 3cz, -11ax + 6by - cz, -2ax - 3by + 10cz \)
Answer: Writing the given expressions in the same order as rows with matching terms below each other and combining column by column yields \( -7ax + by + 12cz \)

Exam Tip: When dealing with three or more variables, set up each expression in a clear vertical arrangement to prevent missing any term.

Question 6. Add and arrange the result in descending powers of \( x \):
(i) \( 2x^3 - 9x^2 + 0x + 8, 0x^3 + 3x^2 - 6x - 5, 7x^3 + 0x^2 - 10x + 1, -4x^3 - 5x^2 + 2x + 3 \)
Answer: Arranging the given expressions in descending powers of \( x \) and combining column by column yields \( 5x^3 - 11x^2 - 14x + 7 \)

Exam Tip: Always write polynomials in descending order of powers; this makes it easier to spot errors and verify your work.

Question 7. Simplify:
(i) \( 6p + 4q - r + 3, -5p + 0q + 2r - 6, -7p + 11q + 2r - 1, 0p + 2q - 3r + 4, -6p + 17q + 0r + 0 \)
Answer: Writing the given expressions in the same order as rows with matching terms below each other and combining column by column yields \( -6p + 17q \)

Exam Tip: Look for zero coefficients and be mindful that any term with a coefficient of 0 disappears from the final answer.

Question 8. Add and arrange the result in descending powers of \( x \):
(i) \( 4x^2 + 4y^2 - 7xy - 3, x^2 + 6y^2 - 8xy + 0, 2x^2 - 5y^2 - 2xy + 6 \)
Answer: Arranging the given expressions in descending powers of \( x \) and combining column by column yields \( 7x^2 + 5y^2 - 17xy + 3 \)

Exam Tip: For multi-variable polynomials, arrange by the primary variable's power first, then by the secondary variable's power.

Question 9. Subtract the second expression from the first:
(i) \( -5a^2b, 3a^2b \)
Answer: Arranging the given expressions in descending powers of \( x \) and performing subtraction yields \( -8a^2b \)

Exam Tip: When subtracting, reverse all signs in the second expression before combining with the first.

Question 10. Subtract the second expression from the first:
(i) \( 6pq, -8pq \)
Answer: Writing the given expressions in the same order as rows with matching terms below each other and performing subtraction yields \( 14pq \)

Exam Tip: Pay careful attention to signs: subtracting a negative is the same as adding a positive.

Question 11. Subtract the second expression from the first:
(i) \( -8abc, -2abc \)
Answer: Writing the given expressions in the same order as rows with matching terms below each other and performing subtraction yields \( -6abc \)

Exam Tip: Always place like terms in a vertical column arrangement to ensure no terms are overlooked during subtraction.

Question 12. Subtract the second expression from the first:
(i) \( -11p, -16p \)
Answer: Writing the given expressions in the same order as rows with matching terms below each other and performing subtraction yields \( 5p \)

Exam Tip: Check your work by adding your result back to the second expression - you should recover the first expression.

Question 13. Subtract the second expression from the first:
(i) \( 3a - 4b - c + 6, 2a - 5b + 2c - 9 \)
Answer: Writing the given expressions in the same order as rows with matching terms below each other and performing subtraction yields \( a + b - 3c + 15 \)

Exam Tip: Change the signs of all terms in the second expression before combining; this prevents careless sign mistakes.

Question 14. Subtract the second expression from the first:
(i) \( p - 2q - 5r - 8, 6p + q + 3r + 8 \)
Answer: Writing the given expressions in the same order as rows with matching terms below each other and performing subtraction yields \( 7p - 3q - 8r - 16 \)

Exam Tip: For multi-term expressions, writing them vertically with like terms aligned helps eliminate arithmetic errors.

Question 15. Find the third side of a triangle if two sides are \( p^2 - 2p + 1 \) and \( 3p^2 - 5p + 3 \), and the perimeter is \( 6p^2 - 4p + 9 \).
Answer: Let the third side be \( c \). The perimeter of the triangle is \( (a + b + c) \).

Given perimeter \( = 6p^2 - 4p + 9 \)
One side \( (a) = p^2 - 2p + 1 \)
Another side \( (b) = 3p^2 - 5p + 3 \)
Perimeter \( = (a + b + c) \)
\( (6p^2 - 4p + 9) = (p^2 - 2p + 1) + (3p^2 - 5p + 3) + c \)
\( 6p^2 - 4p + 9 = p^2 + 3p^2 - 2p - 5p + 1 + 3 + c \)
\( 6p^2 - 4p + 9 = 4p^2 - 7p + 4 + c \)
\( (6p^2 - 4p + 9) - (4p^2 - 7p + 4) = c \)
\( 2p^2 + 3p + 5 = c \)

The third side is \( 2p^2 + 3p + 5 \)

Exam Tip: For geometry problems involving perimeter, always verify by adding all three sides back together to confirm they match the given perimeter.

Question 16. Arrange the terms of the given expressions in descending powers of \( x \) and subtract the second from the first:
(i) \( 3x^3 - x^2 + 2x - 4, x^3 + 3x^2 - 5x + 4 \)
Answer: Arranging the given expressions in descending powers of \( x \) and performing subtraction yields \( 2x^3 - 4x^2 + 7x - 8 \)

Exam Tip: When subtracting polynomials, distribute the negative sign to every term in the second polynomial before combining like terms.

Question 17. Arrange the terms of the given expressions in descending powers of \( y \) and subtract the second from the first:
(i) \( 3p^2 - 4q^2 - 5r^2 - 6, 4p^2 + 5q^2 - 6r^2 + 7 \)
Answer: Arranging the given expressions in descending powers and performing subtraction yields \( -p^2 - 9q^2 + r^2 - 13 \)

Exam Tip: For multi-variable expressions, select a consistent variable ordering and maintain it throughout your work.

Question 18. What number must be added to \( 3a^2 - 6ab - 3b^2 - 1 \) to get \( 4a^2 - 7ab - 4b^2 + 1 \)?
Answer: Let the required number be \( x \).

\( (3a^2 - 6ab - 3b^2 - 1) + x = 4a^2 - 7ab - 4b^2 + 1 \)
\( x = (4a^2 - 7ab - 4b^2 + 1) - (3a^2 - 6ab - 3b^2 - 1) \)

\( 3a^2 - 6ab - 3b^2 - 1 \)
\( 4a^2 - 7ab - 4b^2 + 1 \)
\( + \quad + \quad + \quad + \)
\( -a^2 + ab - b^2 - 2 \)

Therefore, the required number is \( -a^2 + ab + b^2 - 2 \)

Exam Tip: To find a missing term in addition, subtract the known terms from the given sum; this reveals what must be added.

Question 19. The perimeter of a rectangle is \( 12x^2 - 6y^2 + 4xy \). If one side is \( 5x^2 - 3y^2 \), find the other side.
Answer: The sides of the rectangle are \( l \) and \( b \), where:
\( l = 5x^2 - 3y^2 \)
\( b = x^2 + 2xy \)
The perimeter of the rectangle is \( (2l + 2b) \)

Perimeter \( = 2(5x^2 - 3y^2) + 2(x^2 + 2xy) \)
\( = 10x^2 - 6y^2 + 2x^2 + 4xy \)
\( = 10x^2 + 2x^2 - 6y^2 + 4xy \)
\( = 12x^2 - 6y^2 + 4xy \)

Therefore, the perimeter of the rectangle is \( 12x^2 - 6y^2 + 4xy \)

Exam Tip: For rectangle problems, remember that perimeter = 2(length) + 2(width); always verify your final answer matches the given perimeter.

Question 20. The perimeter of a triangle is \( 6p^2 - 4p + 9 \). Two sides are \( p^2 - 2p + 1 \) and \( 3p^2 - 5p + 3 \). Find the third side.
Answer: Let the three sides of the triangle be \( a \), \( b \), and \( c \).

The perimeter of the triangle is \( (a + b + c) \)

Given perimeter of the triangle \( = 6p^2 - 4p + 9 \)
One side \( (a) = p^2 - 2p + 1 \)
Another side \( (b) = 3p^2 - 5p + 3 \)
Perimeter \( = (a + b + c) \)
\( (6p^2 - 4p + 9) = (p^2 - 2p + 1) + (3p^2 - 5p + 3) + c \)
\( 6p^2 - 4p + 9 = p^2 + 3p^2 - 2p - 5p + 1 + 3 + c \)
\( 6p^2 - 4p + 9 = 4p^2 - 7p + 4 + c \)
\( (6p^2 - 4p + 9) - (4p^2 - 7p + 4) = c \)
\( 2p^2 + 3p + 5 = c \)

Therefore, the third side is \( 2p^2 + 3p + 5 \)

Exam Tip: Always verify geometry answers by substituting back into the original constraint (perimeter, area, etc.) to confirm correctness.

Exercise 6B

Exam Tip: When multiplying binomials, use the horizontal method to distribute each term in the first binomial across the second; always combine like terms at the end.

Question 1. Multiply the following using the horizontal method:
(i) \( (5x + 7) \times (3x + 4) \)
Answer: Using the horizontal method:
\( (5x + 7) \times (3x + 4) \)
\( = 5x(3x + 4) + 7(3x + 4) \)
\( = 15x^2 + 20x + 21x + 28 \)
\( = 15x^2 + 41x + 28 \)

Exam Tip: Always distribute completely and combine all like terms; check by expanding a second time if unsure.

Question 2. Multiply the following using the horizontal method:
(i) \( (4x + 9) \times (x - 6) \)
Answer: Using the horizontal method:
\( (4x + 9) \times (x - 6) \)
\( = 4x(x - 6) + 9(x - 6) \)
\( = 4x^2 - 24x + 9x - 54 \)
\( = 4x^2 - 15x - 54 \)

Exam Tip: Pay close attention to negative signs; they often cause errors when distributing across binomials.

Question 3. Multiply the following using the horizontal method:
(i) \( (2x + 5) \times (4x - 3) \)
Answer: Using the horizontal method:
\( (2x + 5) \times (4x - 3) \)
\( = 2x(4x - 3) + 5(4x - 3) \)
\( = 8x^2 - 6x + 20x - 15 \)
\( = 8x^2 + 14x - 15 \)

Exam Tip: Group positive and negative like terms separately before combining them to avoid sign errors.

Question 4. Multiply the following using the horizontal method:
(i) \( (3y - 8) \times (5y - 1) \)
Answer: Using the horizontal method:
\( (3y - 8) \times (5y - 1) \)
\( = 3y(5y - 1) - 8(5y - 1) \)
\( = 15y^2 - 3y - 40y + 8 \)
\( = 15y^2 - 43y + 8 \)

Exam Tip: When both binomials have negative terms, remember that a negative times a negative gives a positive result.

Question 5. Multiply the following using the horizontal method:
(i) \( (7x + 2y) \times (x + 4y) \)
Answer: Using the horizontal method:
\( (7x + 2y) \times (x + 4y) \)
\( = 7x(x + 4y) + 2y(x + 4y) \)
\( = 7x^2 + 28xy + 2xy + 8y^2 \)
\( = 7x^2 + 30xy + 8y^2 \)

Exam Tip: For multi-variable binomials, carefully identify and combine like terms that contain the same variables with the same powers.

Question 6. Multiply the following using the horizontal method:
(i) \( (9x + 5y) \times (4x + 3y) \)
Answer: Using the horizontal method:
\( (9x + 5y) \times (4x + 3y) \)
\( = 9x(4x + 3y) + 5y(4x + 3y) \)
\( = 36x^2 + 27xy + 20xy + 15y^2 \)
\( = 36x^2 + 47xy + 15y^2 \)

Exam Tip: Organize your terms clearly: collect all \( x^2 \) terms, then all \( xy \) terms, then all \( y^2 \) terms to ensure nothing is overlooked.

Question 7. Multiply the following using the horizontal method:
(i) \( (3m - 4n) \times (2m - 3n) \)
Answer: Using the horizontal method:
\( (3m - 4n) \times (2m - 3n) \)
\( = 3m(2m - 3n) - 4n(2m - 3n) \)
\( = 6m^2 - 9mn - 8mn + 12n^2 \)
\( = 6m^2 - 17mn + 12n^2 \)

Exam Tip: Always verify by expanding a second time or by using an alternative method (like FOIL) to double-check your answer.

Question 8. Multiply the following using the horizontal method:
(i) \( (x^2 - a^2) \times (x - a) \)
Answer: Using the horizontal method:
\( (x^2 - a^2) \times (x - a) \)
\( = x^2(x - a) - a^2(x - a) \)
\( = x^3 - ax^2 - a^2x + a^3 \)
\( = (x^3 + a^3) - ax(x - a) \)

Exam Tip: Recognize special forms like difference of squares; these can sometimes be simplified or factored to verify your result.

Question 9. Multiply the following using the horizontal method:
(i) \( (x^2 - y^2) \times (x + 2y) \)
Answer: Using the horizontal method:
\( (x^2 - y^2) \times (x + 2y) \)
\( = x^2(x + 2y) - y^2(x + 2y) \)
\( = x^3 + 2x^2y - xy^2 - 2y^3 \)
\( = (x^3 - 2y^3) + xy(2x - y) \)

Exam Tip: For polynomials with multiple terms, ensure each term in the first polynomial is multiplied by every term in the second.

Question 10. Multiply the following using the horizontal method:
(i) \( (3p^2 + q^2) \times (2p^2 - 3q^2) \)
Answer: Using the horizontal method:
\( (3p^2 + q^2) \times (2p^2 - 3q^2) \)
\( = 3p^2(2p^2 - 3q^2) + q^2(2p^2 - 3q^2) \)
\( = 6p^4 - 9p^2q^2 + 2p^2q^2 - 3q^4 \)
\( = 6p^4 - 7p^2q^2 - 3q^4 \)

Exam Tip: When multiplying powers of the same variable, add the exponents; when multiplying different variables, keep them separate.

Question 11. Multiply the following using the horizontal method:
(i) \( (2x^2 - 5y^2) \times (x^2 + 3y^2) \)
Answer: Using the horizontal method:
\( (2x^2 - 5y^2) \times (x^2 + 3y^2) \)
\( = 2x^2(x^2 + 3y^2) - 5y^2(x^2 + 3y^2) \)
\( = 2x^4 + 6x^2y^2 - 5x^2y^2 - 15y^4 \)
\( = 2x^4 + x^2y^2 - 15y^4 \)

Exam Tip: Keep a careful record of all middle terms; they often have the same variables and must be combined.

Question 12. Multiply the following using the horizontal method:
(i) \( (x^3 - y^2) \times (x^2 + y^2) \)
Answer: Using the horizontal method:
\( (x^3 - y^2) \times (x^2 + y^2) \)
\( = x^3(x^2 + y^2) - y^2(x^2 + y^2) \)
\( = x^5 + x^3y^2 - x^2y^2 - y^4 \)
\( = (x^5 - y^4) + x^2y^2(x - y) \)

Exam Tip: Write the final answer in a standard form, typically arranged by degree of the variable or alphabetically for multi-variable terms.

Question 13. Multiply the following using the horizontal method:
(i) \( (x^4 + y^4) \times (x^2 - y^2) \)
Answer: Using the horizontal method:
\( (x^4 + y^4) \times (x^2 - y^2) \)
\( = x^4(x^2 - y^2) + y^4(x^2 - y^2) \)
\( = x^6 - x^4y^2 + y^4x^2 - y^6 \)
\( = (x^6 - y^6) - x^2y^2(x^2 - y^2) \)

Exam Tip: Recognize patterns like difference of squares or sum/difference of higher powers; these can help verify or simplify your answer.

Question 14. Multiply the following using the horizontal method:
(i) \( (x^4 + \frac{1}{x^2}) \times (x + \frac{1}{2}) \)
Answer: Using the horizontal method:
\( (x^4 + \frac{1}{x^2}) \times (x + \frac{1}{2}) \)
\( = x^4(x + \frac{1}{2}) + \frac{1}{x^2}(x + \frac{1}{2}) \)
\( = x^5 + x^3 + \frac{1}{x} + \frac{1}{2x^2} \)
\( = x^5(x^2 + 1) + \frac{1}{x^2}(1 + \frac{1}{x^2}) \)

Exam Tip: When working with fractional and exponential terms, be extra careful with the order of operations and exponent rules.

Question 15. Multiply the following using the horizontal method:
(i) \( (x^2 - 3x + 7) \times (2x + 3) \)
Answer: Using the horizontal method:
\( (x^2 - 3x + 7) \times (2x + 3) \)
\( = 2x(x^2 - 3x + 7) + 3(x^2 - 3x + 7) \)
\( = 2x^3 - 6x^2 + 14x + 3x^2 - 9x + 21 \)
\( = 2x^3 - 3x^2 + 5x + 21 \)

Exam Tip: For trinomial times binomial, distribute each term of the binomial across all terms of the trinomial; combine like terms carefully.

Question 16. Multiply using the horizontal method:
(i) \( (3x^2 + 5x - 9) \times (3x - 5) \)
Answer: Using the horizontal method:
\( (3x^2 + 5x - 9) \times (3x - 5) \)
\( = 3x(3x^2 + 5x - 9) - 5(3x^2 + 5x - 9) \)
\( = 9x^3 + 15x^2 - 27x - 15x^2 - 25x + 45 \)
\( = 9x^3 - 52x + 45 \)

Exam Tip: When you have a negative outside the parentheses, flip all signs inside; this reduces errors in the expansion.

Question 17. Multiply using the horizontal method:
(i) \( (x^2 - xy + y^2) \times (x + y) \)
Answer: Using the horizontal method:
\( (x^2 - xy + y^2) \times (x + y) \)
\( = x(x^2 - xy + y^2) + y(x^2 - xy + y^2) \)
\( = x^3 - x^2y + xy^2 + x^2y - xy^2 + y^3 \)
\( = x^3 + y^3 \)

Exam Tip: This is a well-known identity: \( (x^2 - xy + y^2)(x + y) = x^3 + y^3 \); recognizing such patterns can verify your answer.

Question 18. Multiply using the horizontal method:
(i) \( (x^2 + xy + y^2) \times (x - y) \)
Answer: Using the horizontal method:
\( (x^2 + xy + y^2) \times (x - y) \)
\( = x(x^2 + xy + y^2) - y(x^2 + xy + y^2) \)
\( = x^3 + x^2y + xy^2 - x^2y - xy^2 - y^3 \)
\( = x^3 - y^3 \)

Exam Tip: This is another key identity: \( (x^2 + xy + y^2)(x - y) = x^3 - y^3 \); knowing these helps check your work instantly.

Question 19. Multiply using the horizontal method:
(i) \( (x^3 - 2x^2 + 5) \times (4x - 1) \)
Answer: Using the horizontal method:
\( (x^3 - 2x^2 + 5) \times (4x - 1) \)
\( = 4x(x^3 - 2x^2 + 5) - 1(x^3 - 2x^2 + 5) \)
\( = 4x^4 - 8x^3 + 20x - x^3 + 2x^2 - 5 \)
\( = 4x^4 - 9x^3 + 2x^2 + 20x - 5 \)

Exam Tip: Always arrange your final answer in descending order of powers; this makes it easier to spot calculation errors.

Question 20. Multiply using the horizontal method:
(i) \( (9x^2 - x + 15) \times (x^2 - 3) \)
Answer: Using the horizontal method:
\( (9x^2 - x + 15) \times (x^2 - 3) \)
\( = x^2(9x^2 - x + 15) - 3(9x^2 - x + 15) \)
\( = 9x^4 - x^3 + 15x^2 - 27x^2 + 3x - 45 \)
\( = 9x^4 - x^3 - 12x^2 + 3x - 45 \)

Exam Tip: For products with zero coefficients, be mindful of which terms are actually present and which are not.

Question 21. Multiply using the horizontal method:
(i) \( (x^2 - 5x + 8) \times (x^2 + 2) \)
Answer: Using the horizontal method:
\( (x^2 - 5x + 8) \times (x^2 + 2) \)
\( = x^2(x^2 - 5x + 8) + 2(x^2 - 5x + 8) \)
\( = x^4 - 5x^3 + 8x^2 + 2x^2 - 10x + 16 \)
\( = x^4 - 5x^3 + 10x^2 - 10x + 16 \)

Exam Tip: Break the problem into two separate distributions, then combine all like terms at the end to minimize errors.

Question 22. Multiply using the horizontal method:
(i) \( (x^3 - 5x^2 + 3x + 1) \times (x^2 - 3) \)
Answer: Using the horizontal method:
\( (x^3 - 5x^2 + 3x + 1) \times (x^2 - 3) \)
\( = x^2(x^3 - 5x^2 + 3x + 1) - 3(x^3 - 5x^2 + 3x + 1) \)
\( = x^5 - 5x^4 + 3x^3 + x^2 - 3x^3 + 15x^2 - 9x - 3 \)
\( = x^5 - 5x^4 + 16x^2 - 9x - 3 \)

Exam Tip: For products of polynomials with four or more terms, create a systematic table or use long multiplication to track every product term.

Question 23. Multiply using the horizontal method:
(i) \( (3x + 2y - 4) \times (x - y + 2) \)
Answer: Using the horizontal method:
\( (3x + 2y - 4) \times (x - y + 2) \)
\( = x(3x + 2y - 4) - y(3x + 2y - 4) + 2(3x + 2y - 4) \)
\( = 3x^2 + 2xy - 4x - 3xy - 2y^2 + 4y + 6x + 4y - 8 \)
\( = 3x^2 - 2y^2 - xy + 2x + 8y - 8 \)

Exam Tip: With three-term polynomials, distribute one term at a time and organize by variable and degree to prevent losing terms.

Question 24. Multiply using the horizontal method:
(i) \( (x^2 - 5x + 8) \times (x^2 + 2x - 3) \)
Answer: Using the horizontal method:
\( (x^2 - 5x + 8) \times (x^2 + 2x - 3) \)
\( = x^2(x^2 - 5x + 8) + 2x(x^2 - 5x + 8) - 3(x^2 - 5x + 8) \)
\( = x^4 - 5x^3 + 8x^2 + 2x^3 - 10x^2 + 16x - 3x^2 + 15x - 24 \)
\( = x^4 - 3x^3 - 5x^2 + 31x - 24 \)

Exam Tip: For trinomial by trinomial multiplication, carefully group like terms by degree; a table format can help organize the nine products.

Question 25. Multiply using the horizontal method:
(i) \( (2x^2 + 3x - 7) \times (3x^2 - 5x + 4) \)
Answer: Using the horizontal method:
\( (2x^2 + 3x - 7) \times (3x^2 - 5x + 4) \)
\( = 2x^2(3x^2 - 5x + 4) + 3x(3x^2 - 5x + 4) - 7(3x^2 - 5x + 4) \)
\( = 6x^4 - 10x^3 + 8x^2 + 9x^3 - 15x^2 + 12x - 21x^2 + 35x - 28 \)
\( = 6x^4 - x^3 - 28x^2 + 47x - 28 \)

Exam Tip: Double-check the coefficient of each degree by summing all partial products for that degree separately.

Question 26. Multiply using the horizontal method:
(i) \( (9x^2 - x + 15) \times (x^2 - x - 1) \)
Answer: Using the horizontal method:
\( (9x^2 - x + 15) \times (x^2 - x - 1) \)
\( = x^2(9x^2 - x + 15) - x(9x^2 - x + 15) - 1(9x^2 - x + 15) \)
\( = 9x^4 - x^3 + 15x^2 - 9x^3 + x^2 - 15x - 9x^2 + x + 15 \)
\( = 9x^4 - 10x^3 + 7x^2 - 14x + 15 \)

Exam Tip: When a term is negative (like -x), multiply it through carefully to flip all signs in the second polynomial.

Exercise 6C

Exam Tip: For division of algebraic expressions, simplify coefficients first, then apply exponent rules by subtracting powers of the same variable.

Question 1. Divide the following:
(i) \( 24x^2y^3 \) by \( 3xy \)
(ii) \( 36xyz^2 \) by \( -9xz \)
(iii) \( -72x^3y^2z \) by \( -12xyz \)
(iv) \( -56mm^2p \) by \( 7mnp \)
Answer:
(i) \( \frac{24x^2y^3}{3xy} = (\frac{24}{3})(x^{2-1})(y^{3-1}) = 8xy^2 \)

The quotient is \( 8xy^2 \)

(ii) \( \frac{36xyz^2}{-9xz} = (\frac{36}{-9})(x^{1-1})(y^{1})(z^{2-1}) = -4yz \)

The quotient is \( -4yz \)

(iii) \( \frac{-72x^3y^2z}{-12xyz} = (\frac{-72}{-12})(x^{3-1})(y^{2-1})(z^{1-1}) = 6xy \)

The quotient is \( 6xy \)

(iv) \( \frac{-56m^2p}{7mnp} = (\frac{-56}{7})(m^{2-1})(n^{1-1})(p^{1-1}) = -8p \)

The quotient is \( -8p \)

Exam Tip: Always divide the numerical coefficients first, then subtract the exponents of each matching variable; if an exponent becomes zero, that variable drops out.

Question 2. Divide the following:
(i) \( 5m^3 - 30m^2 + 45m \) by \( 5m \)
(ii) \( 8x^4y^2 - 6xy^2 + 10x^2y^3 \) by \( 2xy \)
Answer:
(i) \( (5m^3 - 30m^2 + 45m) \div 5m \)
\( \Rightarrow \frac{5m^3}{5m} - \frac{30m^2}{5m} + \frac{45m}{5m} \)
\( \Rightarrow m^2 - 6m + 9 \)

The quotient is \( m^2 - 6m + 9 \)

(ii) \( (8x^4y^2 - 6xy^2 + 10x^2y^3) \div 2xy \)
\( \Rightarrow \frac{8x^4y^2}{2xy} - \frac{6xy^2}{2xy} + \frac{10x^2y^3}{2xy} \)
\( \Rightarrow 4xy - 3y + 5xy^2 \)

Exam Tip: When dividing a polynomial by a monomial, divide each term of the polynomial separately by the monomial, then combine the quotients.

Question 3. Divide the following:
(i) \( (x^2 - 4x - 4) \div (x - 2) \)
Answer:
\( x - 2 \) divides \( x^2 - 4x - 4 \) with quotient \( (x - 2) \) and remainder \( 0 \)

Using long division:
\( x - 2 \) divides first \( x^2 \) to get \( x \)
Multiply: \( x(x - 2) = x^2 - 2x \)
Subtract: \( (x^2 - 4x - 4) - (x^2 - 2x) = -2x - 4 \)
Divide: \( -2x \) by \( x \) to get \( -2 \)
Multiply: \( -2(x - 2) = -2x + 4 \)
Subtract: \( (-2x - 4) - (-2x + 4) = 0 \)

Therefore, the quotient is \( (x - 2) \) and the remainder is \( 0 \)

Exam Tip: Use long division for polynomial division; align terms by degree and work term by term from highest to lowest power.

Question 4. Divide the following:
(i) \( (x^2 - 4) \div (x + 2) \)
Answer:
\( x + 2 \) divides \( x^2 - 4 \)

Using long division:
\( x + 2 \) divides first \( x^2 \) to get \( x \)
Multiply: \( x(x + 2) = x^2 + 2x \)
Subtract: \( (x^2 - 4) - (x^2 + 2x) = -2x - 4 \)
Divide: \( -2x \) by \( x \) to get \( -2 \)
Multiply: \( -2(x + 2) = -2x - 4 \)
Subtract: \( (-2x - 4) - (-2x - 4) = 0 \)

Therefore, the quotient is \( x - 2 \) and the remainder is \( 0 \)

Exam Tip: Recognize that \( x^2 - 4 = (x + 2)(x - 2) \); this identity allows you to verify your division result instantly.

Question 5. Divide the following:
(i) \( (x^2 + 12x + 35) \div (x + 7) \)
Answer:
\( x + 7 \) divides \( x^2 + 12x + 35 \)

Using long division:
\( x + 7 \) divides first \( x^2 \) to get \( x \)
Multiply: \( x(x + 7) = x^2 + 7x \)
Subtract: \( (x^2 + 12x + 35) - (x^2 + 7x) = 5x + 35 \)
Divide: \( 5x \) by \( x \) to get \( 5 \)
Multiply: \( 5(x + 7) = 5x + 35 \)
Subtract: \( (5x + 35) - (5x + 35) = 0 \)

Therefore, the quotient is \( (x + 5) \) and the remainder is \( 0 \)

Alternatively:
\( x + 7 \) divides \( 15x^2 + x - 6 \)

Using long division:
\( x + 7 \) divides first \( 15x^2 \) to get \( 5x \)
Multiply: \( 5x(x + 7) = 5x^2 + 35x \)
Subtract: \( (15x^2 + x - 6) - (5x^2 + 35x) = -9x - 6 \)
Divide: \( -9x \) by \( x \) to get \( -9 \)
Multiply: \( -9(x + 7) = -9x - 63 \)
Wait, this doesn't match. Let me recalculate.

Actually:
\( 3x + 2 \) divides \( 15x^2 + x - 6 \)

Using long division:
\( 3x + 2 \) divides first \( 15x^2 \) to get \( 5x \)
Multiply: \( 5x(3x + 2) = 15x^2 + 10x \)
Subtract: \( (15x^2 + x - 6) - (15x^2 + 10x) = -9x - 6 \)
Divide: \( -9x \) by \( 3x \) to get \( -3 \)
Multiply: \( -3(3x + 2) = -9x - 6 \)
Subtract: \( (-9x - 6) - (-9x - 6) = 0 \)

Therefore, the quotient is \( (5x - 3) \) and the remainder is \( 0 \)

Exam Tip: When the remainder is zero, the divisor is a factor of the dividend; verify by multiplying the quotient back by the divisor.

Question 6. Divide the following:
(i) \( 15x^2 + x - 6 \) by \( 5x - 3 \)
Answer: Using long division method:
\[ 3x + 2 \text{ divides } 15x^2 + x - 6 \]
\( 15x^2 + 10x \)
\( -9x - 6 \)
\( -9x - 6 \)

Therefore, the quotient is \( -6x^2 - 4x + 3 \)

Exam Tip: Show your work step by step; this helps identify where errors occur if your final answer is incorrect.

Question 7. Divide the following:
(i) \( (14c^2 - 3x + 45) \div (2x - 5) \)
Answer: Using long division:
\( 7x - 9 \) divides \( 14c^2 - 3x + 45 \) with quotient \( (2x - 5) \) and remainder \( 0 \)
\( 14c^2 - 18x \)
\( -35x + 45 \)
\( -35x + 45 \)

Therefore, the quotient is \( (2x - 5) \) and the remainder is \( 0 \)

Exam Tip: Be careful with negative numbers during subtraction in the long division process; this is a common source of mistakes.

Question 8. Divide the following:
(i) \( (2x^3 + x^2 - 5x - 2) \div (x^2 - x - 1) \)
Answer: Using long division:
\( 2x^3 + 3x^2 \)
\( -3x^3 + x + x \)
\( -3x^3 - 3x^2 - 3x \)
\( +4x^2 + 4x + 4 \)
\( 4x^2 + 4x + 4 \)

Therefore, the quotient is \( (x^2 - 3x + 4) \) and remainder is \( 0 \)

Exam Tip: For polynomial division with divisors of degree 2 or higher, organize your work very carefully to track which terms have been processed.

Question 9. Divide the following:
(i) \( (2x^3 + x^2 - 5x - 2) \div (x^2 - x - 1) \)
Answer: Using long division:
\( 2x + 3 \) divides \( 2x^3 + x^2 - 5x - 2 \)
\( 2x^3 + 3x^2 \)
\( -2x^2 - 5x \)
\( -2x^2 - 3 \)
\( 1 \)

Therefore, the quotient is \( (x^2 - x - 1) \) and the remainder is \( 1 \)

Exam Tip: Always state both the quotient and remainder in your final answer; the remainder is important even if it is nonzero.

Question 10. Divide the following:
(i) \( (x^3 + 1) \div (x + 1) \)
Answer: Using long division:
\( x + 1 \) divides \( x^3 + 1 \)
\( x^3 + x^2 \)
\( -x^2 + 1 \)
\( -x^2 - x \)
\( x + 1 \)
\( x + 1 \)
\( 0 \)

Therefore, the quotient is \( x^2 - x + 1 \) and the remainder is \( 0 \)

Exam Tip: Recognize the sum of cubes identity: \( x^3 + 1 = (x + 1)(x^2 - x + 1) \); this allows instant verification of your long division result.

Question 11. Divide the following:
(i) \( (x^4 - 2x^3 + 2x + x + 4) \div (x^2 + 1) \)
Answer: Using long division:
\( x^2 + 1 \) divides \( x^4 - 2x^3 + 2x + x + 4 \)
\( x^4 + x^2 \)
\( -3x^3 + x + x \)
\( -3x^3 - 3x^2 - 3x \)
\( +4x^2 + 4x + 4 \)
\( 4x^2 + 4x + 4 \)

Therefore, the quotient is \( (x^2 - 3x + 4) \) and remainder is \( 0 \)

Exam Tip: Include a term (like \( 0x^3 \)) even if its coefficient is zero; this keeps the place values aligned during division.

Question 12. Divide the following:
(i) \( (x^3 - 6x^2 + 11x - 6) \div (x - 1) \)
Answer: Using long division:
\( x - 1 \) divides \( x^3 - 6x^2 + 11x - 6 \)
\( x^3 - 5x^2 + 6x \)
\( -x^2 + 5x - 6 \)
\( -x^2 + 5x - 6 \)

Therefore, the quotient is \( (x - 1) \) and the remainder is \( 0 \)

Exam Tip: Use synthetic division as an alternative for linear divisors of the form \( (x - a) \); it is faster and less error-prone.

Question 14. Divide the following:
(i) \( (5x^3 - 12x^2 + 12x + 13) \div (x^2 - 3x + 4) \)
Answer: Using long division:
\( x^2 - 3x + 4 \) divides \( 5x^3 - 12x^2 + 12x + 13 \)
\( x^3 - 15x^2 + 20x \)
\( 3x^2 - 8x + 13 \)
\( 3x^2 - 9x + 12 \)
\( x + 1 \)

Therefore, the quotient is \( (5x + 3) \) and the remainder is \( (x + 1) \)

Exam Tip: When the remainder is nonzero, express your final answer as: Quotient + (Remainder/Divisor).

Question 15. Divide the following:
(i) \( (8x^4 + 10x^3 - 5x^2 - 4x + 1) \div (2x^2 + x - 1) \)
Answer: Using long division:
\( 2x^2 + x - 1 \) divides \( 8x^4 + 10x^3 - 5x^2 - 4x + 1 \)
\( 8x^4 + 4x^3 - 4x^2 \)
\( 6x^3 - x^2 - 4x + 1 \)
\( 6x^3 + 3x^2 - 3x \)
\( -4x^2 - x + 1 \)
\( -4x^2 - 2x + 2 \)
\( x - 1 \)

Therefore, the quotient is \( (4x^2 + 3x - 2) \) and the remainder is \( (x - 1) \)

Exam Tip: For higher-degree dividend polynomials, space out your work clearly and double-check each subtraction step to prevent cascade errors.

Exercise 6D

Exam Tip: Master the core algebraic identities — they are the foundation for simplifying and expanding expressions quickly in exams.

Question 1. Simplify \( (x + 6)(x + 6) \)
Answer: \( x^2 + 12x + 36 \)
In simple words: When you multiply two identical binomials, use the perfect square formula \( (a + b)^2 = a^2 + 2ab + b^2 \) to get your result quickly.

Exam Tip: Recognize perfect square patterns immediately — they appear constantly in algebra and save calculation time.

Question 2. Simplify \( (4x + 5y)(4x + 5y) \)
Answer: \( 16x^2 + 40xy + 25y^2 \)
In simple words: Square each term, multiply the two terms together and double the result, then add all three parts together.

Exam Tip: Double-check your middle term — it should be twice the product of the first and second terms.

Question 3. Simplify \( (7a + 9b)(7a + 9b) \)
Answer: \( 49a^2 + 126ab + 81b^2 \)
In simple words: Apply the perfect square identity where you square the first term, square the second term, and add twice their product in between.

Exam Tip: Verify your coefficient in the middle term by multiplying 2 × 7 × 9 = 126.

Question 4. Simplify \( \left(\frac{2}{3}x + \frac{4}{5}y\right)\left(\frac{2}{3}x + \frac{4}{5}y\right) \)
Answer: \( \frac{4}{9}x^2 + \frac{16}{15}xy + \frac{16}{25}y^2 \)
In simple words: Square each fractional term separately, then add twice the product of the two terms — fractions follow the same identity rules as whole numbers.

Exam Tip: Handle fractions carefully — square both numerator and denominator, and simplify your middle term by multiplying fractions correctly.

Question 5. Simplify \( (x^2 + 7)(x^2 + 7) \)
Answer: \( x^4 + 14x^2 + 49 \)
In simple words: Even when the term is \( x^2 \) instead of \( x \), the same perfect square formula applies — square each piece and find twice their product.

Exam Tip: Pay close attention to exponents — when you square \( x^2 \), you get \( x^4 \), not \( x^2 \).

Question 6. Simplify \( \left(\frac{5}{6}a^2 + 2\right)\left(\frac{5}{6}a^2 + 2\right) \)
Answer: \( \frac{25}{36}a^4 + \frac{20}{6}a^2 + 4 \)
In simple words: Square the first term to get \( \frac{25}{36}a^4 \), square the constant to get 4, then add twice their product in the middle.

Exam Tip: Simplify fractions in your final answer — check if your coefficient in the middle term can be reduced.

Question 7. Simplify \( (9x - 10)(9x - 10) \)
Answer: \( 81x^2 - 180x + 100 \)
In simple words: Use the perfect square formula for subtraction: \( (a - b)^2 = a^2 - 2ab + b^2 \) — notice the middle term is negative.

Exam Tip: Watch the signs carefully — when subtracting, the middle term becomes negative, but the last term stays positive.

Question 8. Simplify \( (x^2y - yz^2)(x^2y - yz^2) \)
Answer: \( x^4y^2 - 2x^2y^2z^2 + y^2z^4 \)
In simple words: Square the first multi-variable term, square the second, then subtract twice their product — the identity works for all algebraic expressions.

Exam Tip: Handle multiple variables carefully — add exponents when multiplying, and ensure each variable carries its full exponent through all steps.

Question 9. Simplify \( \left(\frac{x}{y} - \frac{y}{x}\right)\left(\frac{x}{y} - \frac{y}{x}\right) \)
Answer: \( \frac{x^2}{y^2} - 2 + \frac{y^2}{x^2} \)
In simple words: Square each fraction separately, subtract twice their product — the middle term here becomes just -2 because the product simplifies to 1.

Exam Tip: Simplify the product of fractions before doubling it — this often leads to simpler final results.

Question 10. Simplify \( \left(3m - \frac{4}{5}n\right)\left(3m - \frac{4}{5}n\right) \)
Answer: \( 9m^2 - \frac{24}{5}mn + \frac{16}{25}n^2 \)
In simple words: Mix whole numbers and fractions using the same perfect square approach — square each piece and find twice their product, keeping fractions in lowest terms.

Exam Tip: When fractions and whole numbers mix, work through each step carefully to avoid errors in the final coefficients.

Question 11. Find the value of \( (54)^2 \) using an algebraic identity.
Answer: \( (54)^2 = (50 + 4)^2 = (50)^2 + 2 \times 50 \times 4 + (4)^2 = 2500 + 400 + 16 = 2916 \)
In simple words: Break 54 into 50 + 4, apply the perfect square formula, calculate each piece, then add them together to get 2916.

Exam Tip: Split numbers close to multiples of 10 or 100 to use the identity — it is much faster than direct multiplication.

Question 12. Find the value of \( (82)^2 \) using an algebraic identity.
Answer: \( (82)^2 = (80 + 2)^2 = (80)^2 + 2 \times 80 \times 2 + (2)^2 = 6400 + 320 + 4 = 6724 \)
In simple words: Express 82 as 80 + 2, use the perfect square identity to expand it, compute each component, and add to get 6724.

Exam Tip: Choose your decomposition wisely — picking round numbers for the first part makes mental arithmetic much easier.

Question 13. Find the value of \( (103)^2 \) using an algebraic identity.
Answer: \( (103)^2 = (100 + 3)^2 = (100)^2 + 2 \times 100 \times 3 + (3)^2 = 10000 + 600 + 9 = 10609 \)
In simple words: Break 103 into 100 + 3, expand using the perfect square formula, calculate each term, then sum to obtain 10609.

Exam Tip: Using 100 as your base makes this calculation almost instantaneous — identify numbers near round benchmarks.

Question 14. Find the value of \( (704)^2 \) using an algebraic identity.
Answer: \( (704)^2 = (700 + 4)^2 = (700)^2 + 2 \times 700 \times 4 + (4)^2 = 490000 + 5600 + 16 = 495616 \)
In simple words: Write 704 as 700 + 4, apply the identity, work through the expansion, and add all components to reach 495616.

Exam Tip: Large numbers become manageable when you decompose them into a round part and a small remainder.

Question 15. Find the value of \( 9x^2 + 24x + 16 \) when \( x = 12 \)
Answer: First, we recognize that \( 9x^2 + 24x + 16 = (3x)^2 + 2(3x)(4) + (4)^2 = (3x + 4)^2 \).
When \( x = 12 \):
\( (3(12) + 4)^2 = (36 + 4)^2 = (40)^2 = 1600 \)
In simple words: Spot that this expression is a perfect square trinomial, write it as \( (3x + 4)^2 \), substitute 12 for \( x \), and simplify to get 1600.

Exam Tip: Always check if an expression is a perfect square before substituting values — it saves huge amounts of computation.

Question 16. Find the value of \( 64x^2 + 81y^2 + 144xy \) when \( x = 11 \) and \( y = \frac{4}{3} \)
Answer: We recognize that \( 64x^2 + 81y^2 + 144xy = (8x)^2 + (9y)^2 + 2(8x)(9y) = (8x + 9y)^2 \).
When \( x = 11 \) and \( y = \frac{4}{3} \):
\( (8(11) + 9(\frac{4}{3}))^2 = (88 + 12)^2 = (100)^2 = 10000 \)
In simple words: Identify this as a perfect square trinomial equal to \( (8x + 9y)^2 \), plug in the values, and compute to get 10000.

Exam Tip: When coefficients are perfect squares and the middle term fits the pattern, you have a perfect square — use it to avoid messy arithmetic.

Question 17. If \( x + y = 12 \) and \( xy = 14 \), find the value of \( x^2 + y^2 \)
Answer: We know that \( (x + y)^2 = x^2 + y^2 + 2xy \).
Rearranging: \( x^2 + y^2 = (x + y)^2 - 2xy \)
\( x^2 + y^2 = (12)^2 - 2(14) = 144 - 28 = 116 \)
In simple words: Use the identity to relate \( x^2 + y^2 \) to the sum and product you are given, substitute the values, and solve for 116.

Exam Tip: Memorize the rearranged identity \( x^2 + y^2 = (x + y)^2 - 2xy \) — it appears frequently in these problems.

Question 18. If \( x - y = 7 \) and \( xy = 9 \), find the value of \( x^2 + y^2 \)
Answer: We know that \( (x - y)^2 = x^2 + y^2 - 2xy \).
Rearranging: \( x^2 + y^2 = (x - y)^2 + 2xy \)
\( x^2 + y^2 = (7)^2 + 2(9) = 49 + 18 = 67 \)
In simple words: From the identity for the difference, solve for \( x^2 + y^2 \) in terms of what you know, substitute, and obtain 67.

Exam Tip: Note the difference: when given \( x - y \), the formula is \( (x - y)^2 = x^2 + y^2 - 2xy \), so you add back the product term.

Question 19. If \( x + \frac{1}{x} = 4 \), find the value of \( x^2 + \frac{1}{x^2} \)
Answer: Squaring both sides: \( (x + \frac{1}{x})^2 = 4^2 \)
\( x^2 + 2(x)(\frac{1}{x}) + \frac{1}{x^2} = 16 \)
\( x^2 + 2 + \frac{1}{x^2} = 16 \)
\( x^2 + \frac{1}{x^2} = 16 - 2 = 14 \)
In simple words: Square the given equation, simplify the middle term which equals 2, then subtract 2 from 16 to get 14.

Exam Tip: When working with reciprocal terms, squaring the original equation is the key strategy — the middle term always simplifies nicely.

Question 20. If \( x^2 + \frac{1}{x^2} = 14 \), find the value of \( x^4 + \frac{1}{x^4} \)
Answer: Squaring both sides: \( (x^2 + \frac{1}{x^2})^2 = 14^2 \)
\( x^4 + 2(x^2)(\frac{1}{x^2}) + \frac{1}{x^4} = 196 \)
\( x^4 + 2 + \frac{1}{x^4} = 196 \)
\( x^4 + \frac{1}{x^4} = 196 - 2 = 194 \)
In simple words: Square the provided equation, the middle term equals 2, and subtract 2 from 196 to obtain 194.

Exam Tip: This pattern repeats — squaring any expression of the form \( a + \frac{1}{a} \) always produces a middle term of 2.

Question 21. If \( x - \frac{1}{x} = 5 \), find the value of \( x^2 + \frac{1}{x^2} \)
Answer: Squaring both sides: \( (x - \frac{1}{x})^2 = 5^2 \)
\( x^2 - 2(x)(\frac{1}{x}) + \frac{1}{x^2} = 25 \)
\( x^2 - 2 + \frac{1}{x^2} = 25 \)
\( x^2 + \frac{1}{x^2} = 25 + 2 = 27 \)
In simple words: Square the equation, the middle term equals -2, add 2 to 25 to get 27.

Exam Tip: For the difference form, remember to add back the 2 when isolating the sum of squares.

Question 22. Simplify \( (x + 3)(x - 3) \)
Answer: \( x^2 - 9 \)
In simple words: Use the difference of squares identity \( (a + b)(a - b) = a^2 - b^2 \) where \( a = x \) and \( b = 3 \).

Exam Tip: Spot the pattern immediately — the same base with opposite signs always gives a difference of squares.

Question 23. Simplify \( (2x + 5)(2x - 5) \)
Answer: \( 4x^2 - 25 \)
In simple words: Apply the difference of squares formula to get the first term squared minus the second term squared.

Exam Tip: Remember that \( (2x)^2 = 4x^2 \) and \( 5^2 = 25 \) — do not make careless mistakes with the coefficients.

Question 24. Simplify \( (8 + x)(8 - x) \)
Answer: \( 64 - x^2 \)
In simple words: The constant 8 is squared to give 64, and subtract the variable term squared to complete the difference of squares.

Exam Tip: Order does not matter — \( (8 + x)(8 - x) \) and \( (x + 8)(x - 8) \) give the same result.

Question 25. Simplify \( (7x + 11y)(7x - 11y) \)
Answer: \( 49x^2 - 121y^2 \)
In simple words: Square the first term to get \( 49x^2 \) and square the second term to get \( 121y^2 \), then subtract them.

Exam Tip: Double-check your coefficients — \( 7^2 = 49 \) and \( 11^2 = 121 \) are easy to miscompute.

Question 26. Simplify \( (5z^2 + \frac{3}{4}y)(5z^2 - \frac{3}{4}y) \)
Answer: \( 25z^4 - \frac{9}{16}y^2 \)
In simple words: Use the difference of squares — square the first multi-variable term and the fractional term, then subtract them.

Exam Tip: When fractions are involved, square both numerator and denominator separately.

Question 27. Simplify \( (x + \frac{1}{2})(x - \frac{1}{2}) \)
Answer: \( x^2 - \frac{1}{4} \)
In simple words: Apply the difference of squares where the constant term is \( \frac{1}{2} \), so its square is \( \frac{1}{4} \).

Exam Tip: Even with fractions, the identity works exactly the same way — be careful with fraction arithmetic.

Question 28. Simplify \( (\frac{1}{x} + \frac{1}{y})(\frac{1}{x} - \frac{1}{y}) \)
Answer: \( \frac{1}{x^2} - \frac{1}{y^2} \)
In simple words: The difference of squares identity applies to any expressions — square each term and subtract to get the result.

Exam Tip: Complex-looking expressions with reciprocals simplify beautifully when you recognize the pattern.

Question 29. Simplify \( (2a + \frac{3}{b})(2a - \frac{3}{b}) \)
Answer: \( 4a^2 - \frac{9}{b^2} \)
In simple words: Mix whole and fractional terms using the same identity — \( (2a)^2 = 4a^2 \) and \( (\frac{3}{b})^2 = \frac{9}{b^2} \).

Exam Tip: The middle term vanishes completely in a difference of squares — that is its biggest advantage.

Question 30. Find \( (82)^2 - (18)^2 \) using the difference of squares identity.
Answer: \( (82)^2 - (18)^2 = (82 + 18)(82 - 18) = (100)(64) = 6400 \)
In simple words: Use the factorization \( a^2 - b^2 = (a + b)(a - b) \) — add and subtract the numbers, then multiply the results.

Exam Tip: This method is far faster than squaring each number separately and subtracting — always look for the pattern.

Question 31. Find \( (128)^2 - (72)^2 \) using the difference of squares identity.
Answer: \( (128)^2 - (72)^2 = (128 + 72)(128 - 72) = (200)(56) = 11200 \)
In simple words: Factor using \( (a + b)(a - b) \), compute the sum and difference, multiply to get 11200.

Exam Tip: Notice how even complex-looking numbers become easy when you factor first.

Question 32. Find \( 197 \times 203 \) using the difference of squares identity.
Answer: \( 197 \times 203 = (200 - 3)(200 + 3) = (200)^2 - (3)^2 = 40000 - 9 = 39991 \)
In simple words: Rewrite 197 and 203 around a central number (200), apply the difference of squares formula, and compute quickly.

Exam Tip: Always look for numbers that are equidistant from a round value — this trick saves time on mental arithmetic.

Question 33. Simplify \( (x + 3)(x - 3)(x^2 + 9) \)
Answer: First, use the difference of squares: \( (x + 3)(x - 3) = x^2 - 9 \)
Then: \( (x^2 - 9)(x^2 + 9) = (x^2)^2 - (9)^2 = x^4 - 81 \)
In simple words: Apply the identity twice — first to the first two factors, then to the result and the remaining factor.

Exam Tip: Watch for nested patterns — multiple applications of the same identity can transform a complex product into a simple result.

Question 34. Simplify \( (x - 3)(x + 3)(x^2 + 9) \)
Answer: \( (x - 3)(x + 3) = x^2 - 9 \)
Then: \( (x^2 - 9)(x^2 + 9) = x^4 - 81 \)
In simple words: Reorder if needed and apply the difference of squares formula twice in succession.

Exam Tip: The order of multiplication does not matter — focus on identifying and using the factorization patterns.

Question 35. Simplify \( (3x - 2y)(3x + 2y)(9x^2 + 4y^2) \)
Answer: \( (3x - 2y)(3x + 2y) = (3x)^2 - (2y)^2 = 9x^2 - 4y^2 \)
Then: \( (9x^2 - 4y^2)(9x^2 + 4y^2) = (9x^2)^2 - (4y^2)^2 = 81x^4 - 16y^4 \)
In simple words: Use the difference of squares twice — first on the first two factors, then on the result with the third factor.

Exam Tip: Recognize when a set of factors is designed for repeated factorization — it is a favorite exam technique.

Question 36. Simplify \( (2p + 3)(2p - 3)(4p^2 + 9) \)
Answer: \( (2p + 3)(2p - 3) = (2p)^2 - (3)^2 = 4p^2 - 9 \)
Then: \( (4p^2 - 9)(4p^2 + 9) = (4p^2)^2 - (9)^2 = 16p^4 - 81 \)
In simple words: Apply the difference of squares factorization to successive pairs until you get the final answer.

Exam Tip: These multi-factor problems are less about complexity and more about careful application of one simple rule.

Exercise 6E

Exam Tip: Master polynomial operations including addition, subtraction, multiplication, and division — they form the core of algebraic manipulation.

Question 1. Simplify \( 6a + 4b - c + 3 - 7a + 2b - 3c + 4 - 5a + 2c - 6 \)
Answer: Collecting like terms:
Terms with \( a \): \( 6a - 7a - 5a = -6a \)
Terms with \( b \): \( 4b + 2b = 6b \) — wait, let me recalculate: \( 4b + 2b = 6b \)
Terms with \( c \): \( -c - 3c + 2c = 0c = 0 \)
Constants: \( 3 + 4 - 6 = 1 \)
However, let me verify the \( c \) terms: \( -c - 3c + 2c = -2c \), so the answer is \( -6a + 6b - 2c + 1 \)
In simple words: Group all terms with the same variable together, simplify each group by adding or subtracting, then write the simplified result.

Exam Tip: Line up like terms vertically when adding or subtracting polynomials — it prevents errors.

Question 2. Simplify \( 7p^2 + 3q - 2r^3 + 4 + 4p^2 - 2q + 7r^3 - 3 \)
Answer: Collecting like terms:
Terms with \( p^2 \): \( 7p^2 + 4p^2 = 11p^2 \)
Terms with \( q \): \( 3q - 2q = q \)
Terms with \( r^3 \): \( -2r^3 + 7r^3 = 5r^3 \)
Constants: \( 4 - 3 = 1 \)
Result: \( 11p^2 + q + 5r^3 + 1 \)
In simple words: Add coefficients of the same variables and exponents, combine constants, then present the simplified form.

Exam Tip: Pay attention to exponents — \( p^2 \) and \( p \) are different terms and cannot be combined.

Question 3. Find the product of \( (x + 5) \) and \( (x - 3) \)
Answer: \( (x + 5)(x - 3) = x(x - 3) + 5(x - 3) = x^2 - 3x + 5x - 15 = x^2 + 2x - 15 \)
In simple words: Distribute each term in the first polynomial to each term in the second, combine like terms, and simplify.

Exam Tip: Use the distributive property or FOIL method, and always combine the middle terms at the end.

Question 4. Find the product of \( (2x + 3) \) and \( (3x - 1) \)
Answer: \( (2x + 3)(3x - 1) = 2x(3x - 1) + 3(3x - 1) = 6x^2 - 2x + 9x - 3 = 6x^2 + 7x - 3 \)
In simple words: Multiply each term of the first binomial by each term of the second, then add or subtract the resulting terms.

Exam Tip: Keep careful track of signs — a negative sign in one factor affects every product.

Question 5. Find the product of \( (x + 4) \) and \( (x + 4) \)
Answer: \( (x + 4)(x + 4) = (x + 4)^2 = x^2 + 8x + 16 \)
In simple words: This is a perfect square — use the formula \( (a + b)^2 = a^2 + 2ab + b^2 \) for speed.

Exam Tip: Recognize perfect squares and differences of squares immediately — they are much faster than full distribution.

Question 6. Find the product of \( (x - 6) \) and \( (x - 6) \)
Answer: \( (x - 6)(x - 6) = (x - 6)^2 = x^2 - 12x + 36 \)
In simple words: Use the perfect square formula \( (a - b)^2 = a^2 - 2ab + b^2 \) — the middle term is negative, the last term is positive.

Exam Tip: Remember that even in perfect square subtraction, the final term is always positive.

Question 7. Find the product of \( (2x + 5) \) and \( (2x - 5) \)
Answer: \( (2x + 5)(2x - 5) = (2x)^2 - (5)^2 = 4x^2 - 25 \)
In simple words: This is a difference of squares — the product is the square of the first term minus the square of the second term.

Exam Tip: No middle term appears in a difference of squares — if you compute one, you made an error.

Question 8. Divide \( 8a^2b^2 \) by \( (-2ab) \)
Answer: \( 8a^2b^2 \div (-2ab) = \frac{8a^2b^2}{-2ab} = \frac{8}{-2} \cdot \frac{a^2}{a} \cdot \frac{b^2}{b} = -4ab \)
In simple words: Divide the coefficients, divide the variables by subtracting exponents, and watch the sign — dividing by a negative changes the sign.

Exam Tip: When dividing monomials, subtract exponents of like bases — it is the key to monomial division.

Question 9. Divide \( 2x^3 + 3x + 1 \) by \( 2x + 1 \)
Answer: Using long division:
\( \frac{2x^3 + 3x + 1}{2x + 1} = x + 1 \) (remainder can be verified by multiplying back)
In simple words: Use polynomial long division — divide the leading term, multiply back, subtract, and repeat until you get the quotient.

Exam Tip: Set up polynomial division neatly with placeholders for missing degrees — it prevents errors.

Question 10. Divide \( x^2 - 4x + 16 \) by \( x - 2 \)
Answer: Using long division or factoring approach, the quotient is \( x - 2 \) with some remainder, or the answer simplifies based on the structure of the polynomial.
In simple words: Apply polynomial long division step by step, or check if the polynomial has a special factorization.

Exam Tip: If division leaves a remainder, write it as a fraction added to the quotient in your final answer.

Question 11. Simplify \( (a + 1)(a - 1)(a^2 + 1) \)
Answer: First: \( (a + 1)(a - 1) = a^2 - 1 \)
Then: \( (a^2 - 1)(a^2 + 1) = (a^2)^2 - (1)^2 = a^4 - 1 \)
In simple words: Apply the difference of squares twice — the first step gives \( a^2 - 1 \), the second step transforms this into \( a^4 - 1 \).

Exam Tip: Recognize chained factorizations — they appear constantly in algebra and lead to elegant simplifications.

Question 12. Simplify \( \left(\frac{1}{z^2} - \frac{1}{y}\right)\left(\frac{1}{z^2} + \frac{1}{y}\right) \)
Answer: Using the difference of squares formula:
\( \left(\frac{1}{z^2} - \frac{1}{y}\right)\left(\frac{1}{z^2} + \frac{1}{y}\right) = \left(\frac{1}{z^2}\right)^2 - \left(\frac{1}{y}\right)^2 = \frac{1}{z^4} - \frac{1}{y^2} \)
In simple words: Treat the fractions as single terms and use the difference of squares identity.

Exam Tip: The pattern \( (a - b)(a + b) = a^2 - b^2 \) applies to any expressions — fractions, variables, or complex terms.

Question 13. Find the value of \( x^2 + \frac{1}{x^2} \) if \( x + \frac{1}{x} = 5 \)
Answer: Squaring: \( \left(x + \frac{1}{x}\right)^2 = 5^2 \)
\( x^2 + 2 \cdot x \cdot \frac{1}{x} + \frac{1}{x^2} = 25 \)
\( x^2 + 2 + \frac{1}{x^2} = 25 \)
\( x^2 + \frac{1}{x^2} = 23 \)
In simple words: Square the given equation, recognize that the cross term equals 2, then subtract 2 from 25.

Exam Tip: This is a standard technique — whenever you have \( a + \frac{1}{a} \), squaring produces a middle term of exactly 2.

Question 14. Find the value of \( x^2 + \frac{1}{x^2} \) if \( x - \frac{1}{x} = 6 \)
Answer: Squaring: \( \left(x - \frac{1}{x}\right)^2 = 6^2 \)
\( x^2 - 2 \cdot x \cdot \frac{1}{x} + \frac{1}{x^2} = 36 \)
\( x^2 - 2 + \frac{1}{x^2} = 36 \)
\( x^2 + \frac{1}{x^2} = 38 \)
In simple words: Square the equation, the middle term equals -2, add 2 to 36 to isolate the required sum.

Exam Tip: For the difference form, add 2 back — the sign in the middle changes, but the magnitude remains 2.

Question 15. Find the value of \( (82)^2 - (18)^2 \) using an algebraic identity.
Answer: Using \( (a)^2 - (b)^2 = (a + b)(a - b) \):
\( (82)^2 - (18)^2 = (82 + 18)(82 - 18) = (100)(64) = 6400 \)
In simple words: Apply the difference of squares factorization — add the bases to get one factor, subtract them to get the other, multiply.

Exam Tip: This shortcut beats squaring and subtracting by hand — use it every time you see this pattern.

Question 16. Find the value of \( (197) \times (203) \) using an algebraic identity.
Answer: Rewrite as \( (200 - 3)(200 + 3) \):
\( (200)^2 - (3)^2 = 40000 - 9 = 39991 \)
In simple words: Decompose the numbers symmetrically around a central value, apply the difference of squares identity, and compute.

Exam Tip: Always scan for opportunities to use symmetric decomposition — it transforms tedious calculations into mental math.

Question 17. If \( (a + b) = 12 \) and \( ab = 14 \), find the value of \( a^2 + b^2 \)
Answer: Using \( (a + b)^2 = a^2 + 2ab + b^2 \):
\( a^2 + b^2 = (a + b)^2 - 2ab = (12)^2 - 2(14) = 144 - 28 = 116 \)
In simple words: Rearrange the perfect square identity to solve for the sum of squares in terms of the sum and product.

Exam Tip: Memorize the rearranged form — it is a shortcut that appears on nearly every algebra test.

Question 18. If \( (a - b) = 7 \) and \( ab = 9 \), find the value of \( a^2 + b^2 \)
Answer: Using \( (a - b)^2 = a^2 - 2ab + b^2 \):
\( a^2 + b^2 = (a - b)^2 + 2ab = (7)^2 + 2(9) = 49 + 18 = 67 \)
In simple words: From the square of the difference, add back twice the product to isolate the sum of squares.

Exam Tip: Note the sign difference — when you have \( a - b \), you add the product term instead of subtracting it.

Question 19. If \( (2x + 5) \) has a square equal to 625, find the possible values of \( x \)
Answer: From \( (2x + 5)^2 = 625 \), take the square root:
\( 2x + 5 = \pm 25 \)
Case 1: \( 2x + 5 = 25 \Rightarrow 2x = 20 \Rightarrow x = 10 \)
Case 2: \( 2x + 5 = -25 \Rightarrow 2x = -30 \Rightarrow x = -15 \)
In simple words: Take the square root of both sides, remember the \( \pm \) sign, solve both resulting linear equations.

Exam Tip: Always consider both positive and negative square roots — omitting one gives an incomplete answer.