CBSE Class 8 Mathematics Factorisation Worksheet Set 02

Question. What are the factors of ax + by + bx + az + ay + bz ?
(A) (bx + ay), (ax + by)
(B) (a + b), (2x + 2y + 2z)
(C) (x + y + z), (a + b)
(D) (x + y- z), (a -b)
Ahswer : C

Question. Find the quotient when 5a²b²c² is divided by 15 abc.
(A) abc/3
(B) 3 abc
(C) 3 a²b²c²
(D) 5 a²b²c²
Ahswer : A

Question. Which of the following statements is correct?
(A) (a - 4) (a - 2) = a² + 8 - 6a
(B) (2p + 3q) (p - q) = 2p² - 3q²
(C) 3p²/3p² = 0
(D) 4 (m - 5) = 4m - 5
Ahswer : A

Question. Which is the correct statement in the following?
(A) (3m+ 4)² = 3m² + 6m + 16
(B) n(3n + 2) = 3n² + 2n
(C) (x - 2)(x - 8) = x² - 16
(D) (p + 2)(p + 4) = p² + 8
Ahswer : B

Question. If (x² + 3x + 5) (x² - 3x + 5) = m² -  n², what is the value of m?
(A) x² - 3x
(B) 3x
(C) x² + 5
(D) x² + 3x
Ahswer : C

Question. Which of the following is one of the factors of x⁴ + 4 ?
(A) x² + 2
(B) x² - 2x + 2
(C) x² - 2
(D) x² + 2x- 2
Ahswer : B

Question. What are the factors of x⁴ + 2x² + 9 ?
(A) (x² + 2x + 3), (x² - 2x + 3)
(B) (x² + 3), (x² - 3)
(C) (x²+ 2x + 3), (x² + 2x + 3)
(D) (x² + 3), (x² + 3)
Ahswer : A

Question. For x² + 2x + 5 to be a factor of x⁴ + px² + q, what must the respective values of p and q be ?
(A) -2 and 5
(B) 5 and 25
(C) 1 0 and 20
(D) 6 and 25
Ahswer : D

Question. What are the factors of x² + xy - 2xz - 2yz ?
(A) (x - y) and (x + 2z)
(B) (x + y) and (x - 2z)
(C) (x - y) and (x - 2z)
(D) (x + y) and (x + 2z)
Ahswer : B

Question. Amrit and Pankaj expanded (x - 5)2. Amrit's answer is x² - 25 and Pankaj's answer is x² - 10x + 25.Which of the following statements is correct ?
(A) Amrit's answer is correct.
(B) Pankaj's answer is wrong.
(C) Both got correct answer.
(D) Pankaj's answer is correct.
Ahswer : D

Question. Divide 6p⁵ + 18p⁴ - 3p² by 3p².
(A) 2p³ - 6p² + 1
(B) 2p³ - 6p² - 1
(C) 2p³ + 6p² - 1
(D) 2p³ + 6p² + 1
Ahswer : C

Question. Find the factors of b² - 7b + 12.
(A) (b- 4), (b- 8)
(B) (b- 3), (b- 4)
(C) (b - 1 0), (b- 1)
(D) (b - 7), (b- 9)
Ahswer : B

Question. Simplify 3lm/8n² x 32ln²/m² ÷ 9ln²/m
(A) 4l/3n
(B) -(4l/3n)
(C) 4n/3l
(D) -4n/3l 
Ahswer : A

Question. Divide q (5q² - 80) by 5q(q + 4).
(A) 5q (q-16)
(B) (q-4)
(C) 5q (q²+ 16)
(D) (q + 16)
Ahswer : B

Question. What are the factors of x⁴ + y⁴ + x²y²?
(A) (x² + y²) and (x² + l-xy)
(B) (x² + y²) and (x² - y²⁾
(C) (x² + y² + xy) and (x² + y² - xy)
(D) Factorisation is not possible. 
Ahswer : C

Question. Choose the factors of 15x² - 26x + 8 from the following.
(A) (3x - 4), (5x + 2)
(B) (3x - 4), (5x- 2)
(C) (3x + 4), (5x- 2)
(D) (3x + 4), (5x + 2)
Ahswer : B

Question. How many factors does (x⁹ - x) have ?
(A) 5
(B) 4
(C) 2
(D) 9
Ahswer : A

Question. Which of the following are the factors of x²/4 - y²/9 ?

Ahswer : C

Question. What is the coefficient of 'a' when 9a² + 18a is divided by (a + 2) ?
(A) 18
(B) 9
(C) 1/2
(D) 2
Ahswer : B

Question. From the following, which are the factors of a² + b - ab -a ?
(A) (a- 1) and (a - b)
(B) (a + b)and(a - 1)
(C) (a + 1)and(a - b)
(D) (a + b) and (a + 1)
Ahswer : A

Question. The expression (p² + 7p + 1 0) is factorised and then divided by (p + 5). What is the quotient?
(A) p- 5
(B) p- 2
(C) p + 2
(D) p + 5
Ahswer : C

Question. Give the factor form of 6~n + 9mn²l + 12mnl.
(A) 3mn(2m + 3nl + 4l)
(B) 4mn(4m + 4nl + 2l)
(C) 5mn(2m + 2nl + l)
(D) 3mn(m + 2nl + 2l)
Ahswer : A

Question. Divide 44(m⁴ - 5m³ - 24m²) by 11 m(m - 8).
(A) 4m²(m - 3)
(B) 4m(m- 3)
(C) 4m²(m +3)
(D) 4m(m +3)
Ahswer : D

Question. Factorise 49x² - 36.
(A) (7x- 6) (7x + 6)
(B) (7x + 36)7
(C) (7x + 6)6
(D) (7x + 6)(7x + 36)
Ahswer : A

Question. Find the factors of 3z² + 9z + 6.
(A) 3(z + 1) (z + 2)
(B) 4(z + 2) (z + 3)
(C) 2(z + 3) (z + 1)
(D) 3(z + 5) (z + 5)
Ahswer : A

Question. The area of a square is (x² + 8x + 16) cm².
Find the length of its side.
(A) (x - 8) cm 
(B) (x - 4) cm
(C) (x + 4) cm
(D) (x + 2) cm
Ahswer : C

Question. Find the factors of 6 mn - 4n + 6 - 9m.
(A) (2m- 1) and (2n - 4)
(B) (4m - 1) and (n- 3)
(C) (3m - 2) and (2n - 3)
(D) (4m - 4) and (n- 1)
Ahswer : C

Question. Which of the following is the exponential form of factors of a² + 4a + 4 ?
(A) (a + 2)²
(B) (a + 1)²
(C) (a- 2)²
(D) (a -1)²
Ahswer : A

Question. What is the result when x³ + 6x² + 9x is factorised ?
(A) x²(x + 2)²
(B) x(x + 3)²
(C) x(x + 3)
(D) x²(x + 3
Ahswer : B

Question. What are the factors of (2x + 3y)² + 2(2x + 3y) (x + y) + (x + y)² ?
(A) (3x + 4y) and (3x - 4y)
(B) (3x + 4y) and (3x + 4y)
(C) (3x + 2y) and (2x - 3y)
(D) (3x - 4y) and (3x + 2y)
Ahswer : B

Question. Which of the following is the correct factorisation ?
(A) 64 - x² = (64 - x) (64 + x)
(B) 27x² - 48 = 3(3x + 4) (3x- 4)
(C) y² - 81 = (y + 9) (y + 9)
(D) 36 - p² = (p - 6) (p + 6)
Ahswer : B

Question. Which of the following factorisations is incorrect?
(A) 200y² - 2 = 2(10y + 1) (10y -1)
(B) 49x² - 36 = (7x + 6) (7x- 6)
(C) 200y² - 2 = 2(10y + 1) (10y + 1)
(D) 36 -100k² = (6 + 10k) (6 -10k)
Ahswer : C

Question. What are the factors of a²b² + c²d - a²c² - b²d² ?
(A) (a + d), (a -d), (b + c) and (b - c)
(B) (a² - b²)
(C) (a² - d²) and (b² + c²)
(D) (a² + d²) and (b² - c²)
Ahswer : A

Question. Divide - 15a³bc³by - 5abc².
(A) 3a¹ c
(B) 3a² c
(C) 3a³ c
(D) 4a² c
Ahswer : B

Question.  Simplify - 3a²b x 15 cab²/-10 cab
(A) (3/2)a²b²
(B) (9/2)a²b²
(C) (3/2)a3b²
(D) (7/2)a²b²
Ahswer : B

Question. Find the quotient of the algebraic terms in 36x³y²z/- 9x²y²z
(A) 4x
(B) 4xy²z
(C) - 4x
(D) 4x³y²z
Ahswer : C

Question. Divide: 4p⁵ - 14p⁴ + 6p³ - 2p² by 2p²
(A) 2p³ + 7p² + 3p + 1
(B) 2p³ - 7p² + 3p- 1
(C) 2p³ -7p² -3p-1
(D) 2p³ +7p² -3p-1
Ahswer : B

Question. Simplify
3(4 + 2m² - m) - 6(3m² + m + 2)/2(2m- 3) + 3(m + 2)
(A) 3(3m + 4)/7
(B) 3(4m + 3)/7
(C) 3(3m - 4)/7
(D) -3(4m + 3)/7
Ahswer : D

Question. What is the factorisation of 3xz- 4yz- 6xp + 8yp?
(A) (3x + 4y) (z + 2p)
(B) (3x + 4y) (z- 2p)
(C) (3x- 4y) (z + 2p)
(D) (3x- 4y) (z- 2p)
Ahswer : D

Question. Give the factor form of 3(2p - q + 4r) + 2p + 15q - 16r.
(A) 4 (2p + 3q + r)
(B) 4 (2p + 3q - r)
(C) 4 (2p- 3q + r)
(D) 4 (2p - 3q - r)
Ahswer : B

Factorise:

Question 1. (i) \(12x + 15\)      (ii) \(14m - 21\)      (iii) \(9n - 12n^2\)
Answer:
(i) \(12x + 15 = 3(4x) + 3(5) = 3(4x + 5)\)
(ii) \(14m - 21 = 7(2m) - 7(3) = 7(2m - 3)\)
(iii) \(9n - 12n^2 = 3n(3) - 3n(4n) = 3n(3 - 4n)\)
In simple words: Find the highest common factor for each term and take it outside the parentheses.

Exam Tip: Always check if the numerical coefficients share a common divisor and if variables have common powers.

Question 2. (i) \(16a^3 - 24ab\)      (ii) \(15ab^2 - 20a^2b\)      (iii) \(12x^2y^3 - 21x^3y^2\)
Answer:
(i) \(16a^3 - 24ab = 8a \cdot 2a^2 - 8a \cdot 3b = 8a(2a^2 - 3b)\)
(ii) \(15ab^2 - 20a^2b = 5ab \cdot 3b - 5ab \cdot 4a = 5ab(3b - 4a)\)
(iii) \(12x^2y^3 - 21x^3y^2 = 3x^2y^2 \cdot 4y - 3x^2y^2 \cdot 7x = 3x^2y^2(4y - 7x)\)
In simple words: Factor out the greatest common divisor of numbers and the lowest power of common variables.

Exam Tip: Be careful with signs and subtraction when factoring out negative quantities.

Question 3. (i) \(24x^3 - 36x^2y\)      (ii) \(10x^3 - 15x^2\)      (iii) \(36x^3y - 60x^2y^3z\)
Answer:
(i) \(24x^3 - 36x^2y = 12x^2(2x) - 12x^2(3y) = 12x^2(2x - 3y)\)
(ii) \(10x^3 - 15x^2 = 5x^2(2x) - 5x^2(3) = 5x^2(2x - 3)\)
(iii) \(36x^3y - 60x^2y^3z = 12x^2y(3x) - 12x^2y(5y^2z) = 12x^2y(3x - 5y^2z)\)
In simple words: Identify the highest common factor for multi-variable terms term by term.

Exam Tip: Check your answer by multiplying the outer factor back into each term inside the bracket.

Question 4. (i) \(9x^3 - 6x^2 + 12x\)      (ii) \(-5 - 10t + 20t^2\)      (iii) \(18a^3b^3 - 27a^2b^3 + 36a^4b^3\)
Answer:
(i) \(9x^3 - 6x^2 + 12x = 3x(3x^2 - 2x + 4)\)
(ii) \(-5 - 10t + 20t^2 = -5(1 + 2t - 4t^2)\)
(iii) \(18a^3b^3 - 27a^2b^3 + 36a^4b^3 = 9a^2b^3(2a - 3 + 4a^2)\)
In simple words: When the first term is negative, factoring out a negative sign helps make the remaining terms positive.

Exam Tip: Ensure all terms inside the parentheses have been divided by the common factor.

Question 5. (i) \(14x^3 + 21x^2y - 28x^2y^2\)      (ii) \(5(x - 4) - 7(x - 4)\)      (iii) \(2m(1 - n) + 3(1 - n)\)
Answer:
(i) \(14x^3 + 21x^2y - 28x^2y^2 = 7x^2(2x + 3y - 4y^2)\)
(ii) \(5(x - 4) - 7(x - 4) = (x - 4)(5 - 7) = -2(x - 4) = 2(4 - x)\)
(iii) \(2m(1 - n) + 3(1 - n) = (1 - n)(2m + 3)\)
In simple words: Treat binomial expressions like \((x - 4)\) or \((1 - n)\) as a single common term.

Exam Tip: Binomial common factors can be factored out just like monomial variables.

Question 6. (i) \(x(x + 3) + 5(x + 3)\)      (ii) \(x^3(2a - b) + x^2(2a - b)\)      (iii) \(9a(3a - 5b) - 12a^2(3a - 5b)\)
Answer:
(i) \(x(x + 3) + 5(x + 3) = (x + 3)(x + 5)\)
(ii) \(x^3(2a - b) + x^2(2a - b) = x^2(2a - b)(x + 1)\)
(iii) \(9a(3a - 5b) - 12a^2(3a - 5b) = 3a(3a - 5b)(3 - 4a)\)
In simple words: Extract the matching binomial parenthesis and then simplify any remaining common factors.

Question 7. \(6a(a - 2b) + 5b(a - 2b)\)      12. \(2a + 6b - 3(a + 3b)^2\)
Answer:
\(6a(a - 2b) + 5b(a - 2b) = (a - 2b)(6a + 5b)\)
\(2a + 6b - 3(a + 3b)^2 = 2(a + 3b) - 3(a + 3b)^2 = (a + 3b)[2 - 3(a + 3b)] = (a + 3b)(2 - 3a - 9b)\)
In simple words: Group common terms and pull out the shared binomial expression.

Exam Tip: Watch out for coefficients that multiply into the entire remaining bracket expression.

Question 8. \(x^3(2a - b) + x^2(2a - b)\)      13. \(12(2x - 3y)^2 - 16(3y - 2x)\)
Answer:
\(x^3(2a - b) + x^2(2a - b) = x^2(2a - b)(x + 1)\)
\(12(2x - 3y)^2 - 16(3y - 2x) = 12(2x - 3y)^2 + 16(2x - 3y) = 4(2x - 3y)[3(2x - 3y) + 4] = 4(2x - 3y)(6x - 9y + 4)\)
In simple words: Note that \((3y - 2x) = -(2x - 3y)\), which allows you to change the sign and combine terms.

Exam Tip: Reversing a subtraction binomial inside a power requires careful sign changes.

Question 9. \((x + 5)^2 - 4(x + 5)\)      14. \(x(a - 3) + y(3 - a)\)
Answer:
\((x + 5)^2 - 4(x + 5) = (x + 5)[(x + 5) - 4] = (x + 5)(x + 1)\)
\(x(a - 3) + y(3 - a) = x(a - 3) - y(a - 3) = (a - 3)(x - y)\)
In simple words: Rewrite \((3 - a)\) as \(-(a - 3)\) to create a common binomial factor.

Exam Tip: Flipping terms in subtraction negates the entire expression outside.

Question 10. \((x + 5)^2 - 4(x + 5)\)      15. \(x^3 - ax - bx + ab\)
Answer:
\((x + 5)^2 - 4(x + 5) = (x + 5)(x + 5 - 4) = (x + 5)(x + 1)\)
\(x^3 - ax - bx + ab = x(x^2 - a) - b(x^2 - a) = (x^2 - a)(x - b)\)
In simple words: Use regrouping (factorisation by grouping) when four terms are present.

Exam Tip: Group terms in pairs so that a common binomial factor emerges.

Question 11. \(3(a - 2b)^2 - 5(a - 2b)\)      16. \(6ab - b^2 + 12ac - 2bc\)
Answer:
\(3(a - 2b)^2 - 5(a - 2b) = (a - 2b)[3(a - 2b) - 5] = (a - 2b)(3a - 6b - 5)\)
\(6ab - b^2 + 12ac - 2bc = b(6a - b) + 2c(6a - b) = (6a - b)(b + 2c)\)
In simple words: Group the first two terms and the last two terms to discover common factors.

Exam Tip: Rearranging terms before grouping can help if common factors do not appear immediately.

Question 12. \(3(a - 2b)^2 - 5(a - 2b)\) (Duplicate index in source)      17. \(ar + br + at + bt\)
Answer:
\(ar + br + at + bt = r(a + b) + t(a + b) = (a + b)(r + t)\)
In simple words: Factor out common variables from the first pair and second pair of terms.

Exam Tip: Grouping works reliably when terms share common letters.

Question 13. \(16(2p - 3q)^2 - 4(2p - 3q)\)      18. \(x^3 - ax - bx + ab\) (Duplicate index in source)
Answer:
\(16(2p - 3q)^2 - 4(2p - 3q) = 4(2p - 3q)[4(2p - 3q) - 1] = 4(2p - 3q)(8p - 12q - 1)\)
\(x^3 - ax - bx + ab = x(x^2 - a) - b(x^2 - a) = (x^2 - a)(x - b)\)
In simple words: Factor out common binomial expressions carefully.

Exam Tip: Verify each factor by expansion.

Question 14. \(x(a - 3) + y(3 - a)\) (Duplicate index in source)      19. \(ab^2 - bc^2 - ab + c^2\)
Answer:
\(x(a - 3) + y(3 - a) = (a - 3)(x - y)\)
\(ab^2 - bc^2 - ab + c^2 = b^2(a - c) - c^2(a - c) \text{ [Wait, let us regroup: } b(ab - c^2) \dots \text{ instead: } a(b^2 - 1) - c^2(b - 1) \dots \text{ Let us regroup as } ab^2 - ab - bc^2 + c^2 = ab(b - 1) - c^2(b - 1) = (b - 1)(ab - c^2)\)
In simple words: Rearrange terms so that common factors are obvious when grouped in pairs.

Exam Tip: Regrouping terms is a powerful technique for polynomials with four or more terms.

Question 15. \((x + y)(2x + 5) - (x + y)(x + 3)\)      20. \(xz - xz + xy - yz\) (Note: simplifying first)
Answer:
\((x + y)(2x + 5) - (x + y)(x + 3) = (x + y)[(2x + 5) - (x + 3)] = (x + y)(2x + 5 - x - 3) = (x + y)(x + 2)\)
\(xz - xz + xy - yz = xy - yz = y(x - z)\)
In simple words: Cancel or combine identical terms before pulling out common factors.

Exam Tip: Always simplify expressions inside brackets before final factorisation.

Question 16. \(16(2p - 3q)^2 - 4(2p - 3q)\)      21. \(ab - by + 12ac - 2bc\) (from 21 in source: \(6ab - b^2 + 12ac - 2bc\))
Answer:
\(16(2p - 3q)^2 - 4(2p - 3q) = 4(2p - 3q)(8p - 12q - 1)\)
\(6ab - b^2 + 12ac - 2bc = b(6a - b) + 2c(6a - b) = (6a - b)(b + 2c)\)
In simple words: Group terms sharing common coefficients and variables.

Exam Tip: Double-check middle terms when expanding grouped factors.

Question 17. \(ar + br + at + bt\)      22. \((ax + by)^2 + (bx - ay)^2\)
Answer:
\(ar + br + at + bt = (a + b)(r + t)\)
\((ax + by)^2 + (bx - ay)^2 = a^2x^2 + 2axby + b^2y^2 + b^2x^2 - 2bxay + a^2y^2 = a^2x^2 + b^2y^2 + b^2x^2 + a^2y^2 = x^2(a^2 + b^2) + y^2(a^2 + b^2) = (a^2 + b^2)(x^2 + y^2)\)
In simple words: Expand squares using algebraic identities and then regroup terms.

Exam Tip: Cross terms \(+2axby\) and \(-2bxay\) cancel each other out completely.

Question 18. \(x^3 - ax - bx + ab\)      23. \(y^2 - xy(1 - x) - x^3\)
Answer:
\(x^3 - ax - bx + ab = (x^2 - a)(x - b)\)
\(y^2 - xy(1 - x) - x^3 = y^2 - xy + x^2y - x^3 = y(y - x) + x^2(y - x) = (y - x)(y + x^2)\)
In simple words: Expand the bracket and rearrange terms to factor by grouping.

Question 19. \(ab^2 - bc^2 - ab + c^2\)      24. \(ab(x^2 + y^2) - xy(a^2 + b^2)\)
Answer:
\(ab^2 - bc^2 - ab + c^2 = (b - 1)(ab - c^2)\)
\(ab(x^2 + y^2) - xy(a^2 + b^2) = abx^2 + aby^2 - a^2xy - b^2xy = ax(bx - ay) + by(ay - bx) = ax(bx - ay) - by(bx - ay) = (bx - ay)(ax - by)\)
In simple words: Expand, regroup carefully, and adjust signs to reveal the common binomial factor.

Exam Tip: Systematic expansion and rearrangement prevent sign errors in complex grouping problems.

Question 20. \((x - 2y)^2 + 4x - 8y\)      25. \(x^3 - 3x^2 + x - 3\)
Answer:
\((x - 2y)^2 + 4x - 8y = (x - 2y)^2 + 4(x - 2y) = (x - 2y)[(x - 2y) + 4] = (x - 2y)(x - 2y + 4)\)
\(x^3 - 3x^2 + x - 3 = x^2(x - 3) + 1(x - 3) = (x - 3)(x^2 + 1)\)
In simple words: Factor out common binomial expressions or group pairs of terms.

Exam Tip: Always write \(+1\) when factoring out the entire remaining group.

Question 21. \(ab^2 - bc^2 - ab + c^2\)      26. \(x^3 - 3x^2 + x - 3\) (Duplicate index)
Answer:
\(ab^2 - bc^2 - ab + c^2 = (b - 1)(ab - c^2)\)
\(x^3 - 3x^2 + x - 3 = (x - 3)(x^2 + 1)\)
In simple words: Group terms pairwise to find common factors.

Exam Tip: Double-check each grouping step for correctness.

Question 22. \(ab^2 - bc^2 - ab + c^2\)      27. \(ab(x^2 + y^2) - xy(a^2 + b^2)\)
Answer:
\(ab^2 - bc^2 - ab + c^2 = (b - 1)(ab - c^2)\)
\(ab(x^2 + y^2) - xy(a^2 + b^2) = (bx - ay)(ax - by)\)
In simple words: Expand and regroup polynomials into factor pairs.

Exam Tip: Look for symmetry in variables \(a, b\) and \(x, y\).

Question 23. \(x^2 - x(a + 2b) + 2ab\)
Answer:
\(x^2 - x(a + 2b) + 2ab = x^2 - ax - 2bx + 2ab = x(x - a) - 2b(x - a) = (x - a)(x - 2b)\)
In simple words: Expand the middle term and factor by grouping.

Exam Tip: Splitting the middle term using the given constants simplifies the factorisation.

Factorise:

Question 1. \(x^2 - 36\)      2. \(4a^2 - 9\)      3. \(81 - 49x^2\)
Answer:
1. \(x^2 - 36 = x^2 - 6^2 = (x - 6)(x + 6)\)
2. \(4a^2 - 9 = (2a)^2 - 3^2 = (2a - 3)(2a + 3)\)
3. \(81 - 49x^2 = 9^2 - (7x)^2 = (9 - 7x)(9 + 7x)\)
In simple words: Use the difference of squares identity: \(a^2 - b^2 = (a - b)(a + b)\).

Exam Tip: Recognise standard square numbers such as 4, 9, 16, 25, 36, 49, 64, 81, 100.

Question 4. \(4x^2 - 9y^2\)      5. \(16a^2 - 225b^2\)      6. \(9a^2b^2 - 25\)
Answer:
4. \(4x^2 - 9y^2 = (2x)^2 - (3y)^2 = (2x - 3y)(2x + 3y)\)
5. \(16a^2 - 225b^2 = (4a)^2 - (15b)^2 = (4a - 15b)(4a + 15b)\)
6. \(9a^2b^2 - 25 = (3ab)^2 - 5^2 = (3ab - 5)(3ab + 5)\)
In simple words: Apply \(a^2 - b^2 = (a - b)(a + b)\) to both numerical and variable terms.

Question 7. \(16a^2 - 144\)      8. \(63a^2 - 112b^2\)      9. \(20a^2 - 45b^2\)
Answer:
7. \(16a^2 - 144 = 16(a^2 - 9) = 16(a - 3)(a + 3)\)
8. \(63a^2 - 112b^2 = 7(9a^2 - 16b^2) = 7(3a - 4b)(3a + 4b)\)
9. \(20a^2 - 45b^2 = 5(4a^2 - 9b^2) = 5(2a - 3b)(2a + 3b)\)
In simple words: Always factor out the greatest common factor first before applying the difference of squares formula.

Exam Tip: Forgetting to take out the common factor first is a very common mistake.

Question 10. \(12x^2 - 27\)      11. \(x^3 - 64x\)      12. \(16x^5 - 144x^3\)
Answer:
10. \(12x^2 - 27 = 3(4x^2 - 9) = 3(2x - 3)(2x + 3)\)
11. \(x^3 - 64x = x(x^2 - 64) = x(x - 8)(x + 8)\)
12. \(16x^5 - 144x^3 = 16x^3(x^2 - 9) = 16x^3(x - 3)(x + 3)\)
In simple words: Factor out \(x\) or \(x^3\) first, then apply the difference of squares identity.

Exam Tip: Continue factoring until no further algebraic identities can be applied.

Question 13. \(3x^5 - 48x^3\)      14. \(16p^3 - 4p\)      15. \(63a^3b^2 - 7\)
Answer:
13. \(3x^5 - 48x^3 = 3x^3(x^2 - 16) = 3x^3(x - 4)(x + 4)\)
14. \(16p^3 - 4p = 4p(4p^2 - 1) = 4p(2p - 1)(2p + 1)\)
15. \(63a^3b^2 - 7 = 7(9a^3b^2 - 1) = 7(3a^{\frac{3}{2}}b - 1)(3a^{\frac{3}{2}}b + 1)\) or left as \(7(9a^3b^2 - 1)\) depending on level.
In simple words: Extract common numerical and variable factors first.

Exam Tip: Ensure all extracted terms are simplified fully.

Question 16. \(1 - (b - c)^2\)      17. \((2a + 3b)^2 - 16c^2\)      18. \((l + m)^2 - (l - m)^2\)
Answer:
16. \(1 - (b - c)^2 = [1 - (b - c)][1 + (b - c)] = (1 - b + c)(1 + b - c)\)
17. \((2a + 3b)^2 - 16c^2 = [(2a + 3b) - 4c][(2a + 3b) + 4c] = (2a + 3b - 4c)(2a + 3b + 4c)\)
18. \((l + m)^2 - (l - m)^2 = [(l + m) - (l - m)][(l + m) + (l - m)] = (l + m - l + m)(l + m + l - m) = (2m)(2l) = 4lm\)
In simple words: Treat entire binomial expressions as single variables in the difference of squares formula.

Exam Tip: Use square brackets when nesting parentheses to avoid sign errors.

Question 19. \((2x + 5y)^2 - 1\)      20. \(36c^2 - (5a + b)^2\)      21. \((3x - 4y)^2 - 25z^2\)
Answer:
19. \((2x + 5y)^2 - 1 = (2x + 5y - 1)(2x + 5y + 1)\)
20. \(36c^2 - (5a + b)^2 = [6c - (5a + b)][6c + (5a + b)] = (6c - 5a - b)(6c + 5a + b)\)
21. \((3x - 4y)^2 - 25z^2 = (3x - 4y - 5z)(3x - 4y + 5z)\)
In simple words: Apply \(A^2 - B^2 = (A - B)(A + B)\) where \(A\) and \(B\) are algebraic expressions.

Question 22. \(x^2 - y^2 - 2y - 1\)      23. \(25 - a^2 - b^2 - 2ab\)      24. \(25a^2 - 4b^2 + 28bc - 49c^2\)
Answer:
22. \(x^2 - y^2 - 2y - 1 = x^2 - (y^2 + 2y + 1) = x^2 - (y + 1)^2 = [x - (y + 1)][x + (y + 1)] = (x - y - 1)(x + y + 1)\)
23. \(25 - a^2 - b^2 - 2ab = 25 - (a^2 + 2ab + b^2) = 5^2 - (a + b)^2 = [5 - (a + b)][5 + (a + b)] = (5 - a - b)(5 + a + b)\)
24. \(25a^2 - 4b^2 + 28bc - 49c^2 = 25a^2 - (4b^2 - 28bc + 49c^2) = (5a)^2 - (2b - 7c)^2 = [5a - (2b - 7c)][5a + (2b - 7c)] = (5a - 2b + 7c)(5a + 2b - 7c)\)
In simple words: Group three terms to form a complete square, then use the difference of squares identity.

Exam Tip: Look for trinomials that form perfect squares like \(a^2 \pm 2ab + b^2\).

Question 25. \(9a^2 - b^2 + 4b - 4\)      26. \(100 - (x - 5)^2\)
Answer:
25. \(9a^2 - b^2 + 4b - 4 = 9a^2 - (b^2 - 4b + 4) = (3a)^2 - (b - 2)^2 = [3a - (b - 2)][3a + (b - 2)] = (3a - b + 2)(3a + b - 2)\)
26. \(100 - (x - 5)^2 = 10^2 - (x - 5)^2 = [10 - (x - 5)][10 + (x - 5)] = (10 - x + 5)(10 + x - 5) = (15 - x)(x + 5)\)
In simple words: Rewrite constant and bracket terms as squares and apply \(A^2 - B^2\).

Exam Tip: Simplify constant terms inside the final parentheses.

Question 27. Evaluate \((405)^2 - (395)^2\)      28. Evaluate \((7.8)^2 - (2.2)^2\)
Answer:
27. \((405)^2 - (395)^2 = (405 - 395)(405 + 395) = (10)(800) = 8000\)
28. \((7.8)^2 - (2.2)^2 = (7.8 - 2.2)(7.8 + 2.2) = (5.6)(10) = 56\)
In simple words: Use the algebraic identity \(a^2 - b^2 = (a - b)(a + b)\) instead of calculating large squares directly.

Exam Tip: This identity saves time and prevents calculation errors with large numbers.

Factorise:

Question 1. \(x^2 + 8x + 16\)      2. \(x^2 + 14x + 49\)      3. \(1 + 2x + x^2\)
Answer:
1. \(x^2 + 8x + 16 = x^2 + 2(x)(4) + 4^2 = (x + 4)^2\)
2. \(x^2 + 14x + 49 = x^2 + 2(x)(7) + 7^2 = (x + 7)^2\)
3. \(1 + 2x + x^2 = 1^2 + 2(1)(x) + x^2 = (1 + x)^2\)
In simple words: Use the perfect square trinomial identity: \(a^2 + 2ab + b^2 = (a + b)^2\).

Exam Tip: Check that the middle term equals twice the product of the square roots of the first and last terms.

Question 4. \(9 + 6z + z^2\)      5. \(x^2 + 6ax + 9a^2\)      6. \(4y^2 + 20y + 25\)
Answer:
4. \(9 + 6z + z^2 = 3^2 + 2(3)(z) + z^2 = (3 + z)^2\)
5. \(x^2 + 6ax + 9a^2 = x^2 + 2(x)(3a) + (3a)^2 = (x + 3a)^2\)
6. \(4y^2 + 20y + 25 = (2y)^2 + 2(2y)(5) + 5^2 = (2y + 5)^2\)
In simple words: Identify the squared terms and verify the middle term matches \(2ab\).

Exam Tip: Always write the final answer in squared binomial form.

Question 7. \(36a^2 + 36a + 9\)      8. \(9m^2 + 24m + 16\)      9. \(z^2 + z + \frac{1}{4}\)
Answer:
7. \(36a^2 + 36a + 9 = (6a)^2 + 2(6a)(3) + 3^2 = (6a + 3)^2 = [3(2a + 1)]^2 = 9(2a + 1)^2\)
8. \(9m^2 + 24m + 16 = (3m)^2 + 2(3m)(4) + 4^2 = (3m + 4)^2\)
9. \(z^2 + z + \frac{1}{4} = z^2 + 2(z)\left(\frac{1}{2}\right) + \left(\frac{1}{2}\right)^2 = \left(z + \frac{1}{2}\right)^2\)
In simple words: Fractions and coefficients follow the exact same perfect square rules.

Exam Tip: Factor out any common numerical factor first if required.

Question 10. \(49a^2 + 84ab + 36b^2\)      11. \(p^2 - 10p + 25\)      12. \(121a^2 - 88ab + 16b^2\)
Answer:
10. \(49a^2 + 84ab + 36b^2 = (7a)^2 + 2(7a)(6b) + (6b)^2 = (7a + 6b)^2\)
11. \(p^2 - 10p + 25 = p^2 - 2(p)(5) + 5^2 = (p - 5)^2\)
12. \(121a^2 - 88ab + 16b^2 = (11a)^2 - 2(11a)(4b) + (4b)^2 = (11a - 4b)^2\)
In simple words: Use \(a^2 - 2ab + b^2 = (a - b)^2\) when the middle term is negative.

Exam Tip: A negative middle term indicates a subtraction inside the binomial square.

Question 13. \(1 - 6x + 9x^2\)      14. \(9y^2 - 12y + 4\)      15. \(16x^2 - 24x + 9\)
Answer:
13. \(1 - 6x + 9x^2 = 1^2 - 2(1)(3x) + (3x)^2 = (1 - 3x)^2\)
14. \(9y^2 - 12y + 4 = (3y)^2 - 2(3y)(2) + 2^2 = (3y - 2)^2\)
15. \(16x^2 - 24x + 9 = (4x)^2 - 2(4x)(3) + 3^2 = (4x - 3)^2\)
In simple words: Match first and last terms to squares and check the middle term.

Question 16. \(m^2 - 4mn + 4n^2\)      17. \(a^2b^2 - 6abc + 9c^2\)      18. \(m^2 + 2m^2n^2 + n^4\) (Note: source reads \(16. m^2 - 4mn + 4n^2\), 17. \(a^2b^2 - 6abc + 9c^2\), 19. \((l + m)^2 - 4lm\))
Answer:
16. \(m^2 - 4mn + 4n^2 = m^2 - 2(m)(2n) + (2n)^2 = (m - 2n)^2\)
17. \(a^2b^2 - 6abc + 9c^2 = (ab)^2 - 2(ab)(3c) + (3c)^2 = (ab - 3c)^2\)
18. For given expression: \((l + m)^2 - 4lm = l^2 + 2lm + m^2 - 4lm = l^2 - 2lm + m^2 = (l - m)^2\)
In simple words: Expand expressions first if necessary, then identify the perfect square pattern.

Exam Tip: Recognizing expanded forms like \(+2lm\) and \(-4lm\) helps simplify expressions quickly.

Factorise:

Question 1. \(x^2 + 5x + 6\)      2. \(y^2 + 10y + 24\)      3. \(z^2 + 12z + 27\)
Answer:
1. \(x^2 + 5x + 6 = x^2 + 2x + 3x + 6 = x(x + 2) + 3(x + 2) = (x + 2)(x + 3)\)
2. \(y^2 + 10y + 24 = y^2 + 6y + 4y + 24 = y(y + 6) + 4(y + 6) = (y + 4)(y + 6)\)
3. \(z^2 + 12z + 27 = z^2 + 9z + 3z + 27 = z(z + 9) + 3(z + 9) = (z + 3)(z + 9)\)
In simple words: Split the middle term into two numbers whose product equals the constant term and whose sum equals the middle coefficient.

Exam Tip: For \(x^2 + bx + c\), find two numbers multiplying to \(c\) and adding to \(b\).

Question 4. \(p^2 + 6p + 8\)      5. \(x^2 + 15x + 56\)      6. \(y^2 + 19y + 60\)
Answer:
4. \(p^2 + 6p + 8 = p^2 + 4p + 2p + 8 = p(p + 4) + 2(p + 4) = (p + 2)(p + 4)\)
5. \(x^2 + 15x + 56 = x^2 + 7x + 8x + 56 = x(x + 7) + 8(x + 7) = (x + 7)(x + 8)\)
6. \(y^2 + 19y + 60 = y^2 + 15y + 4y + 60 = y(y + 15) + 4(y + 15) = (y + 4)(y + 15)\)
In simple words: Test factor pairs of the constant term until the sum matches the middle coefficient.

Exam Tip: List factors systematically if you cannot spot the numbers immediately.

Question 7. \(x^2 + 13x + 40\)      8. \(x^2 + 15x + 56\) (Duplicate)      9. \(p^2 + 6p - 16\)
Answer:
7. \(x^2 + 13x + 40 = x^2 + 8x + 5x + 40 = x(x + 8) + 5(x + 8) = (x + 5)(x + 8)\)
8. \(x^2 + 15x + 56 = (x + 7)(x + 8)\)
9. \(p^2 + 6p - 16 = p^2 + 8p - 2p - 16 = p(p + 8) - 2(p + 8) = (p - 2)(p + 8)\)
In simple words: When the constant is negative, one factor is positive and the other is negative.

Exam Tip: The larger number takes the sign of the middle term.

Question 10. \(x^2 - 10x + 24\)      11. \(x^2 - 23x + 42\)      12. \(z^2 + 122 + 27\) (from 12 in source: \(z^2 + 12z + 27\))
Answer:
10. \(x^2 - 10x + 24 = x^2 - 6x - 4x + 24 = x(x - 6) - 4(x - 6) = (x - 4)(x - 6)\)
11. \(x^2 - 23x + 42 = x^2 - 21x - 2x + 42 = x(x - 21) - 2(x - 21) = (x - 2)(x - 21)\)
12. \(z^2 + 12z + 27 = (z + 3)(z + 9)\)
In simple words: Two negative factors give a positive product and a negative sum.

Exam Tip: Check signs carefully when both factors are negative.

Question 13. \(x^2 - 17x + 16\)      14. \(x^2 - 22x + 117\)      15. \(x^2 - 9x + 20\)
Answer:
13. \(x^2 - 17x + 16 = x^2 - 16x - x + 16 = x(x - 16) - 1(x - 16) = (x - 1)(x - 16)\)
14. \(x^2 - 22x + 117 = x^2 - 13x - 9x + 117 = x(x - 13) - 9(x - 13) = (x - 9)(x - 13)\)
15. \(x^2 - 9x + 20 = x^2 - 5x - 4x + 20 = x(x - 5) - 4(x - 5) = (x - 4)(x - 5)\)
In simple words: Split the middle term into two negative numbers that add to the middle coefficient.

Exam Tip: Verify the product of your two chosen numbers matches the constant term.

Question 16. \(x^2 + x - 132\)      17. \(x^2 + 5x - 104\)      18. \(y^2 + 7y - 144\)
Answer:
16. \(x^2 + x - 132 = x^2 + 12x - 11x - 132 = x(x + 12) - 11(x + 12) = (x - 11)(x + 12)\)
17. \(x^2 + 5x - 104 = x^2 + 13x - 8x - 104 = x(x + 13) - 8(x + 13) = (x - 8)(x + 13)\)
18. \(y^2 + 7y - 144 = y^2 + 16y - 9y - 144 = y(y + 16) - 9(y + 16) = (y - 9)(y + 16)\)
In simple words: When the constant is negative, the larger factor takes the sign of the middle term.

Exam Tip: Ensure the signs of the factors yield the correct middle term sum.

Question 19. \(z^2 + 19z - 150\)      20. \(y^2 + y - 72\)      21. \(a^2 + 6a - 91\)
Answer:
19. \(z^2 + 19z - 150 = z^2 + 25z - 6z - 150 = z(z + 25) - 6(z + 25) = (z - 6)(z + 25)\)
20. \(y^2 + y - 72 = y^2 + 9y - 8y - 72 = y(y + 9) - 8(y + 9) = (y - 8)(y + 9)\)
21. \(a^2 + 6a - 91 = a^2 + 13a - 7a - 91 = a(a + 13) - 7(a + 13) = (a - 7)(a + 13)\)
In simple words: Find two numbers with a difference equal to the middle coefficient and a product equal to the constant.

Exam Tip: Check subtraction signs carefully.

Question 22. \(p^2 - 4p - 77\)      23. \(x^2 - 7x - 30\)      24. \(x^2 - 11x - 42\)
Answer:
22. \(p^2 - 4p - 77 = p^2 - 11p + 7p - 77 = p(p - 11) + 7(p - 11) = (p + 7)(p - 11)\)
23. \(x^2 - 7x - 30 = x^2 - 10x + 3x - 30 = x(x - 10) + 3(x - 10) = (x + 3)(x - 10)\)
24. \(x^2 - 11x - 42 = x^2 - 14x + 3x - 42 = x(x - 14) + 3(x - 14) = (x + 3)(x - 14)\)
In simple words: Split the middle term so that the negative factor is larger in magnitude.

Exam Tip: The sign of the middle term dictates the sign of the larger factor.

Question 25. \(x^2 - 5x - 24\)      26. \(y^2 - 6y - 135\)      27. \(z^2 - 12z - 45\)
Answer:
25. \(x^2 - 5x - 24 = x^2 - 8x + 3x - 24 = x(x - 8) + 3(x - 8) = (x + 3)(x - 8)\)
26. \(y^2 - 6y - 135 = y^2 - 15y + 9y - 135 = y(y - 15) + 9(y - 15) = (y + 9)(y - 15)\)
27. \(z^2 - 12z - 45 = z^2 - 15z + 3z - 45 = z(z - 15) + 3(z - 15) = (z + 3)(z - 15)\)
In simple words: Find factors of the constant whose difference equals the middle coefficient.

Exam Tip: Verify both the sum and product after splitting.

Question 28. \(x^2 - 4x - 12\)      29. \(3x^2 + 10x + 8\)      30. \(3y^2 + 14y + 8\)
Answer:
28. \(x^2 - 4x - 12 = x^2 - 6x + 2x - 12 = x(x - 6) + 2(x - 6) = (x + 2)(x - 6)\)
29. \(3x^2 + 10x + 8 = 3x^2 + 6x + 4x + 8 = 3x(x + 2) + 4(x + 2) = (3x + 4)(x + 2)\)
30. \(3y^2 + 14y + 8 = 3y^2 + 12y + 2y + 8 = 3y(y + 4) + 2(y + 4) = (3y + 2)(y + 4)\)
In simple words: When the leading coefficient is not 1, multiply the leading coefficient by the constant term before splitting the middle term.

Exam Tip: This is known as splitting the middle term for quadratic polynomials with leading coefficients greater than 1.

Question 31. \(3z^2 - 10z + 8\)      32. \(2x^2 + x - 45\)      33. \(6p^2 + 11p - 10\)
Answer:
31. \(3z^2 - 10z + 8 = 3z^2 - 6z - 4z + 8 = 3z(z - 2) - 4(z - 2) = (3z - 4)(z - 2)\)
32. \(2x^2 + x - 45 = 2x^2 + 10x - 9x - 45 = 2x(x + 5) - 9(x + 5) = (2x - 9)(x + 5)\)
33. \(6p^2 + 11p - 10 = 6p^2 + 15p - 4p - 10 = 3p(2p + 5) - 2(2p + 5) = (3p - 2)(2p + 5)\)
In simple words: Find two numbers whose product equals the product of leading coefficient and constant, and whose sum equals the middle coefficient.

Exam Tip: Grouping terms correctly after splitting is crucial for accuracy.

Question 34. \(2x^2 - 17x - 30\)      35. \(7y^2 - 19y - 6\)      36. \(28 - 31x - 5x^2\)
Answer:
34. \(2x^2 - 17x - 30 = 2x^2 - 20x + 3x - 30 = 2x(x - 10) + 3(x - 10) = (2x + 3)(x - 10)\)
35. \(7y^2 - 19y - 6 = 7y^2 - 21y + 2y - 6 = 7y(y - 3) + 2(y - 3) = (7y + 2)(y - 3)\)
36. \(28 - 31x - 5x^2 = -5x^2 - 31x + 28 = -(5x^2 + 31x - 28) = -(5x^2 + 35x - 4x - 28) = -[5x(x + 7) - 4(x + 7)] = -(5x - 4)(x + 7) = (4 - 5x)(x + 7)\)
In simple words: Arrange terms in descending powers of the variable before factoring.

Exam Tip: Rearranging into standard form \(ax^2 + bx + c\) prevents sign confusion.

Question 37. \(3 + 23z - 8z^2\)      38. \(6x^2 - 5x - 6\)      39. \(3m^2 + 24m + 36\)
Answer:
37. \(3 + 23z - 8z^2 = -8z^2 + 23z + 3 = -(8z^2 - 23z - 3) = -(8z^2 - 24z + z - 3) = -[8z(z - 3) + 1(z - 3)] = -(8z + 1)(z - 3) = (3 - z)(8z + 1) \text{ or } (1 + 8z)(3 - z)\)
38. \(6x^2 - 5x - 6 = 6x^2 - 9x + 4x - 6 = 3x(2x - 3) + 2(2x - 3) = (3x + 2)(2x - 3)\)
39. \(3m^2 + 24m + 36 = 3(m^2 + 8m + 12) = 3(m^2 + 6m + 2m + 12) = 3[m(m + 6) + 2(m + 6)] = 3(m + 2)(m + 6)\)
In simple words: Extract common factors first, then split the middle term.

Exam Tip: Factoring out common constants simplifies the numbers inside the quadratic expression.

Question 40. \(4n^2 - 8n + 3\)      41. \(6x^2 - 17x - 3\)      42. \(7x^2 - 19x - 6\)
Answer:
40. \(4n^2 - 8n + 3 = 4n^2 - 6n - 2n + 3 = 2n(2n - 3) - 1(2n - 3) = (2n - 1)(2n - 3)\)
41. \(6x^2 - 17x - 3 = 6x^2 - 18x + x - 3 = 6x(x - 3) + 1(x - 3) = (6x + 1)(x - 3)\)
42. \(7x^2 - 19x - 6 = 7x^2 - 21x + 2x - 6 = 7x(x - 3) + 2(x - 3) = (7x + 2)(x - 3)\)
In simple words: Split the middle term using the product of the leading coefficient and constant term.

Exam Tip: Check your factorisation by multiplying the binomial factors back together.

OBJECTIVE QUESTIONS

Tick (\(\checkmark\)) the correct answer in each of the following:

Question 1. \((7a^2 - 63b^2) = ?\)
(a) \((7a - 9b)(9a + 7b)\)
(b) \((7a - 9b)(7a + 9b)\)
(c) \(9(a - 3b)(a + 3b)\)
(d) \(7(a - 3b)(a + 3b)\)
Answer: (d) \(7(a - 3b)(a + 3b)\)
\(7a^2 - 63b^2 = 7(a^2 - 9b^2) = 7[a^2 - (3b)^2] = 7(a - 3b)(a + 3b)\)
In simple words: Factor out 7 first, then apply the difference of squares identity.

Exam Tip: Look for common factors before applying formulas.

Question 2. \((2x - 32x^3) = ?\)
(a) \(2(x - 4)(x + 4)\)
(b) \(2x(1 - 2x)^2\)
(c) \(2x(1 + 2x)^2\)
(d) \(2x(1 - 4x)(1 + 4x)\)
Answer: (d) \(2x(1 - 4x)(1 + 4x)\)
\(2x - 32x^3 = 2x(1 - 16x^2) = 2x[1^2 - (4x)^2] = 2x(1 - 4x)(1 + 4x)\)
In simple words: Factor out \(2x\) to leave a difference of squares.

Exam Tip: Always extract the highest common monomial factor first.

Question 3. \(x^3 - 144x = ?\)
(a) \(x(x - 12)^2\)
(b) \(x(x + 12)^2\)
(c) \(x(x - 12)(x + 12)\)
(d) None of the options
Answer: (c) \(x(x - 12)(x + 12)\)
\(x^3 - 144x = x(x^2 - 144) = x(x^2 - 12^2) = x(x - 12)(x + 12)\)
In simple words: Factor out \(x\) and use \(a^2 - b^2\).

Exam Tip: 144 is the square of 12.

Question 4. \((2 - 50x^2) = ?\)
(a) \(2(1 - 5x)^2\)
(b) \(2(1 + 5x)^2\)
(c) \((2 - 5x)(2 + 5x)\)
(d) \(2(1 - 5x)(1 + 5x)\)
Answer: (d) \(2(1 - 5x)(1 + 5x)\)
\(2 - 50x^2 = 2(1 - 25x^2) = 2[1^2 - (5x)^2] = 2(1 - 5x)(1 + 5x)\)
In simple words: Factor out 2 and apply the difference of squares formula.

Exam Tip: Do not confuse a difference of squares with a square of a binomial.

Question 5. \(a^2 + bc + ab + ac = ?\)
(a) \((a + b)(a + c)\)
(b) \((a + b)(b + c)\)
(c) \((b + c)(c + a)\)
(d) \(a(a + b + c)\)
Answer: (a) \((a + b)(a + c)\)
\(a^2 + bc + ab + ac = a^2 + ab + ac + bc = a(a + b) + c(a + b) = (a + b)(a + c)\)
In simple words: Rearrange and group terms to factor by grouping.

Exam Tip: Rearranging terms helps when common factors are not immediately adjacent.

Question 6. \(pq^2 + q(p - 1) - 1 = ?\)
(a) \((pq + 1)(q - 1)\)
(b) \(p(q + 1)(q - 1)\)
(c) \(q(p - 1)(q + 1)\)
(d) \((pq - 1)(q + 1)\)
Answer: (d) \((pq - 1)(q + 1)\)
\(pq^2 + q(p - 1) - 1 = pq^2 + pq - q - 1 = pq(q + 1) - 1(q + 1) = (pq - 1)(q + 1)\)
In simple words: Expand the bracket and group terms.

Exam Tip: Expansion followed by grouping is a standard technique for complex expressions.

Question 7. \(ab - mn + an - bm = ?\)
(a) \((a - b)(m - n)\)
(b) \((a - m)(b + n)\)
(c) \((a - n)(m + b)\)
(d) \((m - a)(n - b)\)
Answer: (a) \((a - b)(m - n)\) (Note: After proper grouping \(ab + an - bm - mn = a(b + n) - m(b + n) = (a - m)(b + n) \dots\) Let us re-verify: } \(ab - mn + an - bm = ab + an - bm - mn = a(b + n) - m(b + n) = (a - m)(b + n) \text{ which corresponds to option (b).}\)
Answer: (b) \((a - m)(b + n)\)
In simple words: Rearrange terms to group \(a\) terms and \(m\) terms.

Exam Tip: Regrouping terms with common variables makes factoring straightforward.

Question 8. \(ab - a - b + 1 = ?\)
(a) \((a - 1)(b - 1)\)
(b) \((1 - a)(1 - b)\)
(c) \((a - 1)(1 - b)\)
(d) \((1 - a)(b - 1)\)
Answer: (a) \((a - 1)(b - 1)\)
\(ab - a - b + 1 = a(b - 1) - 1(b - 1) = (a - 1)(b - 1)\)
In simple words: Group the first two terms and the last two terms.

Exam Tip: Factor out \(-1\) carefully from the last two terms.

Question 9. \(x^2 - xz + xy - yz = ?\)
(a) \((x - y)(x + z)\)
(b) \((x - y)(x - z)\)
(c) \((x + y)(x - z)\)
(d) \((x - y)(z - x)\)
Answer: (c) \((x + y)(x - z)\)
\(x^2 - xz + xy - yz = x(x - z) + y(x - z) = (x + y)(x - z)\)
In simple words: Group terms pairwise to reveal the common binomial factor \((x - z)\).

Exam Tip: Grouping terms directly without rearrangement works when terms are already ordered appropriately.