Factorization

Q1: 7x² – 63y²

Step 1: Take out the common factor
\(7x^2 – 63y^2 = 7(x^2 – 9y^2)\)
Step 2: Factorize the difference of squares
\(x^2 – 9y^2 = (x – 3y)(x + 3y)\)
Step 3: Combine the factor of 7
\(7(x^2 – 9y^2) = 7(x – 3y)(x + 3y)\)
Answer: d. 7(x – 3y)(x + 3y)

Q2: 2p – 32p³

Step 1: Find the common factor of both terms. \[ = 2p(1 – 16p^2) \]Step 2: Recognize that \(1 – 16p^2\) is a difference of squares: \[ 1 – 16p^2 = (1 – 4p)(1 + 4p) \]Step 3: Substitute back into the expression: \[ = 2p(1 – 4p)(1 + 4p) \]Final Answer: \[ {2p(1 – 4p)(1 + 4p)} \]Answer: c. 2p(1 – 4p)(1 + 4p)

Q3: a³ – 144a

Step 1: Take out the common factor
\(a^3 – 144a = a(a^2 – 144)\)
Step 2: Factorize the difference of squares
\(a^2 – 144 = (a – 12)(a + 12)\)
Step 3: Combine the factors
\(a^3 – 144a = a(a – 12)(a + 12)\)
Answer: b. a(a – 12)(a + 12)

Q4: 2 – 50z²

Step 1: Take out the common factor
\(2 – 50z^2 = 2(1 – 25z^2)\)
Step 2: Factorize the difference of squares
\(1 – 25z^2 = (1 – 5z)(1 + 5z)\)
Step 3: Combine the factor of 2
\(2(1 – 25z^2) = 2(1 – 5z)(1 + 5z)\)
Answer: c. 2(1 – 5z)(1 + 5z)

Q5: x² + yz + xy + xz

Step 1: Group the terms
\(x^2 + xy + xz + yz = (x^2 + xy) + (xz + yz)\)
Step 2: Factor each group
\((x^2 + xy) + (xz + yz) = x(x + y) + z(x + y)\)
Step 3: Factor out the common binomial factor
\(x(x + y) + z(x + y) = (x + y)(x + z)\)
Answer: d. (x + y)(x + z)

Q6: ab² + b(a – 1) – 1

Step 1: Expand b(a – 1)
\(ab^2 + b(a – 1) – 1 = ab^2 + ab – b – 1\)
Step 2: Group the terms
\((ab^2 + ab) – (b + 1)\)
Step 3: Factor each group
\(ab(b + 1) – 1(b + 1)\)
Step 4: Factor out the common binomial factor
\((ab – 1)(b + 1)\)
Answer: a. (ab – 1)(b + 1)

Q7: xy – pq + xq – yp

Step 1: Group the terms
\(xy – pq + xq – yp = (xy + xq) – (yp + pq)\)
Step 2: Factor each group
\(x(y + q) – p(y + q)\)
Step 3: Factor out the common binomial factor
\((x – p)(y + q)\)
Answer: c. (x – p)(y + q)

Q8: mn – m – n + 1

Step 1: Group the terms
\(mn – m – n + 1 = (mn – m) – (n – 1)\)
Step 2: Factor each group
\(m(n – 1) – 1(n – 1)\)
Step 3: Factor out the common binomial factor
\((m – 1)(n – 1)\)
Answer: d. (m – 1)(n – 1)

Q9: a² – ac + ab – bc

Step 1: Group the terms
\(a^2 – ac + ab – bc = (a^2 – ac) + (ab – bc)\)
Step 2: Factor each group
\(a(a – c) + b(a – c)\)
Step 3: Factor out the common binomial factor
\((a + b)(a – c)\)
Answer: b. (a + b)(a – c)

Q10: 12a² – 27

Step 1: Take out the common factor
\(12a^2 – 27 = 3(4a^2 – 9)\)
Step 2: Factorize the difference of squares
\(4a^2 – 9 = (2a – 3)(2a + 3)\)
Step 3: Combine with the common factor
\(12a^2 – 27 = 3(2a – 3)(2a + 3)\)
Answer: a. 3(2a – 3)(2a + 3)

Q11: y³ – y

Step 1: Take out the common factor
\(y^3 – y = y(y^2 – 1)\)
Step 2: Factorize the difference of squares
\(y^2 – 1 = (y – 1)(y + 1)\)
Step 3: Combine the factors
\(y^3 – y = y(y – 1)(y + 1)\)
Answer: d. y(y – 1)(y + 1)

Q12: 1 – 2xy – (x² + y²)

Step 1: Rewrite the expression
\(1 – 2xy – (x^2 + y^2) = 1 – x^2 – 2xy – y^2\)
Step 2: Factor by recognizing a perfect square
\(1 – x^2 – 2xy – y^2 = -((x + y)^2 – 1)\)
Step 3: Apply difference of squares
\(-((x + y)^2 – 1) = (1 – (x + y))(1 + (x + y)) = (1 – x – y)(1 + x + y)\)
Answer: b. (1 + x + y)(1 – x – y)

Q13: a² + 6a + 8

Step 1: Identify the coefficients
\(a^2 + 6a + 8 \Rightarrow \text{Coefficient of } a^2 = 1, \text{coefficient of } a = 6, \text{constant} = 8\)
Step 2: Find two numbers whose product is 8 and sum is 6
\(2 \times 4 = 8 \quad \text{and} \quad 2 + 4 = 6\)
Step 3: Factorize the trinomial
\(a^2 + 6a + 8 = (a + 2)(a + 4)\)
Answer: b. (a + 2)(a + 4)

Q14: p² + 4p – 21

i. Step-by-step solution

Step 1: Identify the coefficients
\(p^2 + 4p – 21 \Rightarrow \text{Coefficient of } p^2 = 1, \text{coefficient of } p = 4, \text{constant} = -21\)
Step 2: Find two numbers whose product is -21 and sum is 4
\(7 \times (-3) = -21 \quad \text{and} \quad 7 + (-3) = 4\)
Step 3: Factorize the trinomial
\(p^2 + 4p – 21 = (p + 7)(p – 3)\)
Answer: c. (p + 7)(p – 3)

Q15: x² + 2x – 3

Step 1: Identify the coefficients
\(x^2 + 2x – 3 \Rightarrow \text{Coefficient of } x^2 = 1, \text{coefficient of } x = 2, \text{constant} = -3\)
Step 2: Find two numbers whose product = -3 and sum = 2
\(3 \times (-1) = -3 \quad \text{and} \quad 3 + (-1) = 2\)
Step 3: Factorize the trinomial
\(x^2 + 2x – 3 = (x + 3)(x – 1)\)
Answer: d. (x + 3)(x – 1)

Q16: 40 + 3y – y²

Step 1: Rewrite in standard quadratic form
\(40 + 3y – y^2 = -y^2 + 3y + 40 = -(y^2 – 3y – 40)\)
Step 2: Find two numbers whose product = -40 and sum = -3
\(8 \times (-5) = -40 \quad \text{and} \quad 5 + (-8) = -3\)
Step 3: Factorize
\(-(y^2 – 3y – 40) = -(y + 5)(y – 8) = (5 + y)(8 – y)\)
Answer: b. (5 + y)(8 – y)

Q17: 2a² + 5a + 3

i. Step-by-step solution

Step 1: Identify coefficients
\(2a^2 + 5a + 3 \Rightarrow \text{Coefficient of } a^2 = 2, \text{coefficient of } a = 5, \text{constant} = 3\)
Step 2: Multiply coefficient of a² and constant
\(2 \times 3 = 6\)
Find two numbers whose product = 6 and sum = 5
\(2 \times 3 = 6 \quad \text{and} \quad 2 + 3 = 5\)
Step 3: Split the middle term
\(2a^2 + 2a + 3a + 3\)
Group terms: \((2a^2 + 2a) + (3a + 3)\)
Factor each group: \(2a(a + 1) + 3(a + 1)\)
Factor out common factor: \((2a + 3)(a + 1)\)
Answer: c. (a + 1)(2a + 3)

Q18: 6x² – 13x + 6

Step 1: Identify coefficients
\(6x^2 – 13x + 6 \Rightarrow \text{Coefficient of } x^2 = 6, \text{coefficient of } x = -13, \text{constant} = 6\)
Step 2: Multiply coefficient of x² and constant
\(6 \times 6 = 36\)
Find two numbers whose product = 36 and sum = -13
\(-9 \text{ and } -4\)
Step 3: Split the middle term
\(6x^2 – 9x – 4x + 6\)
Group terms: \((6x^2 – 9x) + (-4x + 6)\)
Factor each group: \(3x(2x – 3) – 2(2x – 3)\)
Factor out common factor: \((3x – 2)(2x – 3)\)
Answer: d. (3x – 2)(2x – 3)

Q19: 4p² – 8p + 3

i. Step-by-step solution

Step 1: Identify coefficients
\(4p^2 – 8p + 3 \Rightarrow \text{Coefficient of } p^2 = 4, \text{coefficient of } p = -8, \text{constant} = 3\)
Step 2: Multiply coefficient of p² and constant
\(4 \times 3 = 12\)
Find two numbers whose product = 12 and sum = -8
\(-6 \text{ and } -2\)
Step 3: Split the middle term
\(4p^2 – 6p – 2p + 3\)
Group terms: \((4p^2 – 6p) + (-2p + 3)\)
Factor each group: \(2p(2p – 3) – 1(2p – 3)\)
Factor out common factor: \((2p – 1)(2p – 3)\)
Answer: d. (2p – 1)(2p – 3)

Q20: 3 + 23x – 8x²

Step 1: Arrange in standard form
\(-8x^2 + 23x + 3 \Rightarrow 8x^2 – 23x – 3 \ (\text{multiply by -1})\)
Step 2: Multiply coefficient of x² and constant
\(8 \times (-3) = -24\)
Find two numbers whose product = -24 and sum = -23
\(-24 \text{ and } 1\)
Step 3: Split middle term
\(8x^2 – 24x + x – 3\)
Group terms: \((8x^2 – 24x) + (x – 3)\)
Factor each group: \(8x(x – 3) + 1(x – 3)\)
Factor out common factor: \((8x + 1)(x – 3)\)
Multiply back by -1: \(-(8x + 1)(x – 3) = (1 + 8x)(3 – x)\)
Answer: c. (1 + 8x)(3 – x)

Consider the following four algebraic expressions:
A. \(x^2+x-12\)
B. \(x^2-9\)
C. \(x^2-8x+16\)
D. \(x^2+6x+9\)

Q21: Which of the following of expressions do not have any common factor?

Step 1: Factorize all expressions:
A: \(x^2 + x – 12 = (x + 4)(x – 3)\)
B: \(x^2 – 9 = (x – 3)(x + 3)\)
C: \(x^2 – 8x + 16 = (x – 4)^2\)
D: \(x^2 + 6x + 9 = (x + 3)^2\)
Step 2: Check common factors between pairs:
– A & B: Common factor = \(x – 3\) → they do have a common factor
– A & C: Factors of A = \((x+4),(x-3)\), Factors of C = \((x-4)\) → no common factor
– B & D: Factors of B = \((x-3),(x+3)\), Factors of D = \((x+3)\) → common factor = \(x+3\)
Answer: b. A and C

Q22: \((x-3)\) is a factor of how many of the given expressions?

Step 1: Check which expressions have \(x-3\) as a factor:
– A: \((x + 4)(x – 3)\) → yes
– B: \((x – 3)(x + 3)\) → yes
– C: \((x – 4)^2\) → no
– D: \((x + 3)^2\) → no
Step 2: Count → 2 expressions (A and B)
Answer: c. Two

Q23: Expression A shares a common factor with which of the following expressions?

Given:
Expression A = \(x^2 + x – 12\)
Step 1: Factorize Expression A:
\(x^2 + x – 12 = (x + 4)(x – 3)\)
Step 2: Factorize each option:
a. \(x^2 – 2x – 15 = (x – 5)(x + 3)\)
b. \(x^2 – 9x + 20 = (x – 4)(x – 5)\)
c. \(x^2 + x – 20 = (x + 5)(x – 4)\)
d. \(x^2 – 16 = (x – 4)(x + 4)\)
Step 3: Find common factors with A:
Factors of A → \((x + 4)\) and \((x – 3)\)
Compare with each option:
a. \((x – 5)(x + 3)\): no common factor
b. \((x – 4)(x – 5)\): no common factor
c. \((x + 5)(x – 4)\): no common factor
d. \((x – 4)(x + 4)\): common factor \((x + 4)\)
Answer:d. \(x^2 – 16\)

Q24: Expression C shares a common factor with which of the following expressions?

Given:
Expression C = \(x^2 – 8x + 16\)
Step 1: Factorize Expression C:
\(x^2 – 8x + 16 = (x – 4)(x – 4) = (x – 4)^2\)
Step 2: Factorize each option:
a. \(x^2 + 2x + 8\): Does not factor into real linear factors.
b. \(x^2 – 7x + 12 = (x – 3)(x – 4)\)
c. \(x^2 – x – 20 = (x – 5)(x + 4)\)
d. \(x^2 – 3x – 4 = (x – 4)(x + 1)\)
Step 3: Find common factors with C:
Factors of C → \((x – 4)\) twice.
Compare with each option:
a. No common factor
b. \((x – 3)(x – 4)\): common factor \((x – 4)\)
c. \((x – 5)(x + 4)\): no common factor
d. \((x – 4)(x + 1)\): common factor \((x – 4)\)
Step 4: Conclusion:
Expression C shares a common factor with options b and d.
Answer:b. \(x^2 – 7x + 12\) and d. \(x^2 – 3x – 4\)