Class 8 Maths - Factorisation NCERT Solutions

Chapter 14: Factorisation (NCERT Solutions)

Exercise 14.1

Q1. Find the common factors of the given terms.
(i) 12x, 36
(ii) 2y, 22xy
(iii) 14pq, 28p²q²
(iv) 2x, 3x², 4
(v) 6abc, 24ab², 12a²b
(vi) 16x³, -4x², 32x
(vii) 10pq, 20qr, 30rp
(viii) 3x²y³, 10x³y², 6x²y²z

(i) 12x = 2×2×3×x. 36 = 2×2×3×3.
Common factors: 2, 2, 3. Highest Common Factor = 12.

(ii) 2y = 2×y. 22xy = 2×11×x×y.
Common Factor = 2y.

(iii) 14pq = 2×7×p×q. 28p²q² = 2×2×7×p×p×q×q.
Common Factor = 14pq.

(iv) 2x = 2×x. 3x² = 3×x×x. 4 = 2×2.
Common Factor = 1.

(v) 6abc = 2×3×a×b×c. 24ab² = 2×2×2×3×a×b×b. 12a²b = 2×2×3×a×a×b.
Common Factor = 6ab.

(vi) 16x³, -4x², 32x = 2×2×2×2×x×x×x, -1×2×2×x×x, 2×2×2×2×2×x.
Common Factor = 2×2×x = 4x.

(vii) 10pq, 20qr, 30rp = 2×5×p×q, 2×2×5×q×r, 2×3×5×r×p.
Common Factor = 2×5 = 10.

(viii) Common Factor = x²y².

Q2. Factorise the following expressions.
(i) 7x - 42
(ii) 6p - 12q
(iii) 7a² + 14a
(iv) -16z + 20z³
(v) 20l²m + 30alm
(vi) 5x²y - 15xy²
(vii) 10a² - 15b² + 20c²
(viii) -4a² + 4ab - 4ca
(ix) x²yz + xy²z + xyz²
(x) ax²y + bxy² + cxyz

(i) 7(x - 6)

(ii) 6(p - 2q)

(iii) 7a(a + 2)

(iv) 4z(-4 + 5z²)

(v) 10lm(2l + 3a)

(vi) 5xy(x - 3y)

(vii) 5(2a² - 3b² + 4c²)

(viii) -4a(a - b + c)

(ix) xyz(x + y + z)

(x) xy(ax + by + cz)

Q3. Factorise.
(i) x² + xy + 8x + 8y
(ii) 15xy - 6x + 5y - 2
(iii) ax + bx - ay - by
(iv) 15pq + 15 + 9q + 25p
(v) z - 7 + 7xy - xyz

(i) x(x + y) + 8(x + y) = (x + y)(x + 8).

(ii) 3x(5y - 2) + 1(5y - 2) = (5y - 2)(3x + 1).

(iii) x(a + b) - y(a + b) = (a + b)(x - y).

(iv) Rearrange: 15pq + 25p + 9q + 15 = 5p(3q + 5) + 3(3q + 5) = (3q + 5)(5p + 3).

(v) z(1 - xy) - 7(1 - xy) = (1 - xy)(z - 7).

Exercise 14.2

Q1. Factorise following expressions:
(i) a² + 8a + 16
(ii) p² - 10p + 25
(iii) 25m² + 30m + 9
(iv) 49y² + 84yz + 36z²
(v) 4x² - 8x + 4
(vi) 121b² - 88bc + 16c²
(vii) (l + m)² - 4lm
(viii) a⁴ + 2a²b² + b⁴

(i) (a)² + 2(a)(4) + (4)² = (a + 4)².

(ii) (p)² - 2(p)(5) + (5)² = (p - 5)².

(iii) (5m)² + 2(5m)(3) + (3)² = (5m + 3)².

(iv) (7y)² + 2(7y)(6z) + (6z)² = (7y + 6z)².

(v) (2x)² - 2(2x)(2) + (2)² = (2x - 2)² = 4(x - 1)².

(vi) (11b)² - 2(11b)(4c) + (4c)² = (11b - 4c)².

(vii) l² + 2lm + m² - 4lm = l² - 2lm + m² = (l - m)².

(viii) (a²)² + 2(a²)(b²) + (b²)² = (a² + b²)².

Q2. Factorise:
(i) 4p² - 9q²
(ii) 63a² - 112b²
(iii) 49x² - 36
(iv) 16x⁵ - 144x³
(v) (l+m)² - (l-m)²
(vi) 9x²y² - 16
(vii) (x² - 2xy + y²) - z²
(viii) 25a² - 4b² + 28bc - 49c²

(i) (2p)² - (3q)² = (2p - 3q)(2p + 3q).

(ii) 7(9a² - 16b²) = 7[(3a)² - (4b)²] = 7(3a - 4b)(3a + 4b).

(iii) (7x)² - (6)² = (7x - 6)(7x + 6).

(iv) 16x³(x² - 9) = 16x³(x - 3)(x + 3).

(v) [(l+m) - (l-m)][(l+m) + (l-m)] = [2m][2l] = 4lm.

(vi) (3xy)² - (4)² = (3xy - 4)(3xy + 4).

(vii) (x - y)² - z² = (x - y - z)(x - y + z).

(viii) 25a² - (4b² - 28bc + 49c²) = (5a)² - (2b - 7c)² = (5a - 2b + 7c)(5a + 2b - 7c).

Q3. Factorise the expressions:
(i) ax² + bx
(ii) 7p² + 21q²
(iii) 2x³ + 2xy² + 2xz²
(iv) am² + bm² + bn² + an²
(v) (lm + l) + m + 1
(vi) y(y + z) + 9(y + z)
(vii) 5y² - 20y - 8z + 2yz
(viii) 10ab + 4a + 5b + 2
(ix) 6xy - 4y + 6 - 9x

(i) x(ax + b)

(ii) 7(p² + 3q²)

(iii) 2x(x² + y² + z²)

(iv) m²(a + b) + n²(b + a) = (a + b)(m² + n²).

(v) l(m + 1) + 1(m + 1) = (m + 1)(l + 1).

(vi) (y + z)(y + 9)

(vii) 5y(y - 4) + 2z(y - 4) = (y - 4)(5y + 2z).

(viii) 2a(5b + 2) + 1(5b + 2) = (5b + 2)(2a + 1).

(ix) 2y(3x - 2) - 3(3x - 2) = (3x - 2)(2y - 3).

Q4. Factorise:
(i) a⁴ - b⁴
(ii) p⁴ - 81
(iii) x⁴ - (y+z)⁴
(iv) x⁴ - (x-z)⁴
(v) a⁴ - 2a²b² + b⁴

(i) (a²)² - (b²)² = (a² - b²)(a² + b²) = (a - b)(a + b)(a² + b²).

(ii) (p²)² - (9)² = (p² - 9)(p² + 9) = (p - 3)(p + 3)(p² + 9).

(iii) [x² - (y+z)²][x² + (y+z)²] = [x - (y+z)][x + (y+z)][x² + (y+z)²] = (x - y - z)(x + y + z)(x² + (y+z)²).

(iv) [x² - (x-z)²][x² + (x-z)²] = [x - (x-z)][x + x - z][x² + x² - 2xz + z²] = z(2x - z)(2x² - 2xz + z²).

(v) (a² - b²)² = [(a - b)(a + b)]² = (a - b)² (a + b)².

Q5. Factorise the following expressions:
(i) p² + 6p + 8
(ii) q² - 10q + 21
(iii) p² + 6p - 16

(i) p² + 4p + 2p + 8 = p(p + 4) + 2(p + 4) = (p + 4)(p + 2).

(ii) q² - 7q - 3q + 21 = q(q - 7) - 3(q - 7) = (q - 7)(q - 3).

(iii) p² + 8p - 2p - 16 = p(p + 8) - 2(p + 8) = (p + 8)(p - 2).

Exercise 14.3

Q1. Carry out the following divisions.
(i) 28x⁴ ÷ 56x
(ii) -36y³ ÷ 9y²
(iii) 66pq²r³ ÷ 11qr²
(iv) 34x³y³z³ ÷ 51xy²z³
(v) 12a⁸b⁸ ÷ (-6a⁶b⁴)

(i) 28/56 × x^4-1 = x³ / 2.

(ii) -36/9 × y^3-2 = -4y.

(iii) 66/11 × p × q^2-1 × r^3-2 = 6pqr.

(iv) 34/51 × x^3-1 × y^3-2 × z^3-3 = (2/3)x²y.

(v) 12/(-6) × a^8-6 × b^8-4 = -2a²b⁴.

Q2. Divide the given polynomial by the given monomial.
(i) (5x² - 6x) ÷ 3x
(ii) (3y⁸ - 4y⁶ + 5y⁴) ÷ y⁴
(iii) 8(x³y²z² + x²y³z² + x²y²z³) ÷ 4x²y²z²
(iv) (x³ + 2x² + 3x) ÷ 2x
(v) (p³q⁶ - p⁶q³) ÷ p³q³

(i) (5x² / 3x) - (6x / 3x) = (5x / 3) - 2.

(ii) (3y⁸ / y⁴) - (4y⁶ / y⁴) + (5y⁴ / y⁴) = 3y⁴ - 4y² + 5.

(iii) (8x²y²z²(x + y + z)) / (4x²y²z²) = 2(x + y + z).

(iv) x(x² + 2x + 3) / 2x = (1/2)(x² + 2x + 3).

(v) p³q³(q³ - p³) / p³q³ = q³ - p³.

Q3. Work out the following divisions.
(i) (10x - 25) ÷ 5
(ii) (10x - 25) ÷ (2x - 5)
(iii) 10y(6y + 21) ÷ 5(2y + 7)
(iv) 9x²y²(3z - 24) ÷ 27xy(z - 8)
(v) 96abc(3a - 12)(5b - 30) ÷ 144(a - 4)(b - 6)

(i) 5(2x - 5) / 5 = 2x - 5.

(ii) 5(2x - 5) / (2x - 5) = 5.

(iii) 10y × 3(2y + 7) / 5(2y + 7) = (30y) / 5 = 6y.

(iv) 9x²y² × 3(z - 8) / 27xy(z - 8) = 27x²y² / 27xy = xy.

(v) 96abc × 3(a - 4) × 5(b - 6) / 144(a - 4)(b - 6) = (1440abc) / 144 = 10abc.

Q4. Divide as directed.
(i) 5(2x + 1)(3x + 5) ÷ (2x + 1)
(ii) 26xy(x + 5)(y - 4) ÷ 13x(y - 4)
(iii) 52pqr(p + q)(q + r)(r + p) ÷ 104pq(q + r)(r + p)
(iv) 20(y + 4)(y² + 5y + 3) ÷ 5(y + 4)
(v) x(x + 1)(x + 2)(x + 3) ÷ x(x + 1)

(i) 5(3x + 5).

(ii) 2y(x + 5).

(iii) (1/2)r(p + q).

(iv) 4(y² + 5y + 3).

(v) (x + 2)(x + 3).

Q5. Factorise the expressions and divide them as directed.
(i) (y² + 7y + 10) ÷ (y + 5)
(ii) (m² - 14m - 32) ÷ (m + 2)
(iii) (5p² - 25p + 20) ÷ (p - 1)
(iv) 4yz(z² + 6z - 16) ÷ 2y(z + 8)
(v) 5pq(p² - q²) ÷ 2p(p + q)
(vi) 12xy(9x² - 16y²) ÷ 4xy(3x + 4y)
(vii) 39y³(50y² - 98) ÷ 26y²(5y + 7)

(i) (y + 5)(y + 2) / (y + 5) = y + 2.

(ii) (m - 16)(m + 2) / (m + 2) = m - 16.

(iii) 5(p² - 5p + 4) / (p - 1) = 5(p - 4)(p - 1) / (p - 1) = 5(p - 4).

(iv) 4yz(z + 8)(z - 2) / 2y(z + 8) = 2z(z - 2).

(v) 5pq(p - q)(p + q) / 2p(p + q) = (5/2)q(p - q).

(vi) 12xy(3x - 4y)(3x + 4y) / 4xy(3x + 4y) = 3(3x - 4y).

(vii) 39y³ × 2(25y² - 49) / 26y²(5y + 7) = 78y³(5y - 7)(5y + 7) / 26y²(5y + 7) = 3y(5y - 7).

Exercise 14.4

Find and correct the errors in the following mathematical statements:

1. 4(x - 5) = 4x - 5
Correct: 4(x - 5) = 4x - 20

2. x(3x + 2) = 3x² + 2
Correct: x(3x + 2) = 3x² + 2x

3. 2x + 3y = 5xy
Correct: 2x + 3y = 2x + 3y (These are unlike terms, cannot be added)

4. x + 2x + 3x = 5x
Correct: x + 2x + 3x = 6x

5. 5y + 2y + y - 7y = 0
Correct: 5y + 2y + y - 7y = 8y - 7y = y

6. 3x + 2x = 5x²
Correct: 3x + 2x = 5x

7. (2x)² + 4(2x) + 7 = 2x² + 8x + 7
Correct: (2x)² + 4(2x) + 7 = 4x² + 8x + 7

8. (2x)² + 5x = 4x + 5x = 9x
Correct: (2x)² + 5x = 4x² + 5x

9. (3x + 2)² = 3x² + 6x + 4
Correct: (3x + 2)² = 9x² + 12x + 4

10. Substituting x = -3 in (x² + 5x + 4) gives (-3)² + 5(-3) + 4 = 9 + 2 + 4 = 15
Correct: (-3)² + 5(-3) + 4 = 9 - 15 + 4 = -2

11. (y - 3)² = y² - 9
Correct: (y - 3)² = y² - 6y + 9

12. (z + 5)² = z² + 25
Correct: (z + 5)² = z² + 10z + 25

13. (2a + 3b)(a - b) = 2a² - 3b²
Correct: (2a + 3b)(a - b) = 2a² - 2ab + 3ab - 3b² = 2a² + ab - 3b²

14. (a + 4)(a + 2) = a² + 8
Correct: (a + 4)(a + 2) = a² + 2a + 4a + 8 = a² + 6a + 8

15. (a - 4)(a - 2) = a² - 8
Correct: (a - 4)(a - 2) = a² - 2a - 4a + 8 = a² - 6a + 8

16. 3x² / 3x² = 0
Correct: 3x² / 3x² = 1

17. (3x² + 1) / 3x² = 1 + 1 = 2
Correct: (3x² + 1) / 3x² = (3x² / 3x²) + (1 / 3x²) = 1 + (1 / 3x²)

18. 3x / (3x + 2) = 1/2
Correct: 3x / (3x + 2) cannot be simplified further.

19. 3 / (4x + 3) = 1 / 4x
Correct: 3 / (4x + 3) cannot be simplified further.

20. (4x + 5) / 4x = 5
Correct: (4x + 5) / 4x = (4x / 4x) + (5 / 4x) = 1 + (5 / 4x)

21. (7x + 5) / 5 = 7x
Correct: (7x + 5) / 5 = (7x / 5) + (5 / 5) = (7x / 5) + 1

Class 8 Maths - Factorisation Practice Questions

Chapter 14: Factorisation (Practice Questions)

RD Sharma / Extra Practice Questions

Q1. Factorise: 12x²y - 18xy²

The greatest common factor of 12 and 18 is 6.
The highest common variables are xy.
Taking out 6xy common:
= 6xy(2x - 3y).

Q2. Factorise: am + bm + an + bn

Group the terms: (am + bm) + (an + bn)
Take 'm' common from first group, 'n' common from second:
= m(a + b) + n(a + b)
Now (a + b) is common:
= (a + b)(m + n).

Q3. Factorise: x² + 8x + 16

This is of the form a² + 2ab + b² = (a + b)².
Here, a = x, b = 4. 2ab = 2(x)(4) = 8x.
So, x² + 8x + 16 = (x + 4)².

Q4. Factorise: 25p² - 49q²

This is of the form a² - b² = (a - b)(a + b).
25p² = (5p)² and 49q² = (7q)².
= (5p - 7q)(5p + 7q).

Q5. Factorise: y² - 7y + 12

Find two numbers whose product is 12 and sum is -7.
The numbers are -3 and -4.
y² - 3y - 4y + 12 = y(y - 3) - 4(y - 3)
= (y - 3)(y - 4).

Q6. Divide: 24(x²yz + xy²z + xyz²) by 8xyz

Take xyz common from the polynomial inside bracket:
24xyz(x + y + z) / 8xyz
Cancel out xyz and divide 24 by 8:
= 3(x + y + z).

Q7. Factorise: 16x⁵ - 144x³

Take 16x³ common:
16x³(x² - 9)
Now apply a² - b² on (x² - 3²):
= 16x³(x - 3)(x + 3).

Q8. Factorise: l²m + lm² + l + m

Group terms: (l²m + lm²) + (l + m)
Take lm common from first group:
lm(l + m) + 1(l + m)
Now take (l + m) common:
= (l + m)(lm + 1).

Q9. Factorise: p² + 6p - 16

Find two numbers whose product is -16 and sum is 6.
The numbers are +8 and -2.
p² + 8p - 2p - 16 = p(p + 8) - 2(p + 8)
= (p + 8)(p - 2).

Q10. Divide: (10y - 25) by (2y - 5)

Factorise the numerator by taking 5 common:
10y - 25 = 5(2y - 5).
Now, 5(2y - 5) / (2y - 5)
Cancel out (2y - 5).
= 5.

Q11. Factorise: 49x² - 84xy + 36y²

This is of the form a² - 2ab + b².
a = 7x, b = 6y. 2ab = 2(7x)(6y) = 84xy.
= (7x - 6y)².

Q12. Divide: y(5y² - 80) by 5y(y + 4)

Numerator = y × 5(y² - 16) = 5y(y - 4)(y + 4)
Denominator = 5y(y + 4)
Cancel out 5y and (y + 4):
= y - 4.

Q13. Factorise: x⁴ - 81

x⁴ - 81 = (x²)² - (9)²
= (x² - 9)(x² + 9)
Further factorise (x² - 9) as (x - 3)(x + 3).
= (x - 3)(x + 3)(x² + 9).

Q14. Find the correct mathematical expression for (3x + 2)²

Using (a + b)² = a² + 2ab + b².
(3x)² + 2(3x)(2) + 2²
= 9x² + 12x + 4.

Q15. Factorise: z - 7 + 7xy - xyz

Rearrange: (z - xyz) - (7 - 7xy)
Take z common from first, -7 common from second:
z(1 - xy) - 7(1 - xy)
= (1 - xy)(z - 7).

Q16. Factorise: a² - 2ab + b² - c²

Recognize the first three terms as a perfect square:
(a - b)² - c²
Now apply x² - y² = (x - y)(x + y):
= (a - b - c)(a - b + c).

Q17. Evaluate using identities: 98² - 2²

Using a² - b² = (a - b)(a + b)
= (98 - 2)(98 + 2)
= 96 × 100
= 9600.

Q18. Factorise: m² - 256

m² - (16)²
= (m - 16)(m + 16).

Q19. Divide: 44(x⁴ - 5x³ - 24x²) by 11x(x - 8)

Numerator: 44x²(x² - 5x - 24)
Factorise x² - 5x - 24 = (x - 8)(x + 3).
So, Numerator = 44x²(x - 8)(x + 3).
Divide by 11x(x - 8).
Cancel 11x and (x - 8):
= 4x(x + 3) = 4x² + 12x.

Q20. Factorise: 3x² + 9x + 6

Take 3 common first: 3(x² + 3x + 2).
Factorise x² + 3x + 2 = (x + 2)(x + 1).
= 3(x + 1)(x + 2).

Class 8 Maths - Factorisation Summary

Chapter 14: Factorisation (Concepts & Formulas)

1. What is Factorisation?

Factorisation is the reverse process of multiplication.
When an algebraic expression is written as the product of its factors, it is called factorisation. The factors can be numbers, algebraic variables, or algebraic expressions.

An irreducible factor is a factor which cannot be expressed further as a product of factors.

2. Methods of Factorisation

Method of Common Factors: Find the common factors present in all terms of the expression and take them out.
E.g., 2x + 4 = 2(x + 2)
Factorisation by Regrouping Terms: Group the terms such that a common factor can be found in each group, and then factorise further.
E.g., 2xy + 2y + 3x + 3 = 2y(x + 1) + 3(x + 1) = (x + 1)(2y + 3)
Factorisation using Identities: Use standard algebraic identities to factorise expressions.
Factorisation of the form x² + (a+b)x + ab: Find two numbers a and b such that their sum equals the coefficient of x, and their product equals the constant term.
x² + (a+b)x + ab = (x + a)(x + b)

3. Standard Algebraic Identities

These identities are crucial for quick factorisation:

(a + b)² = a² + 2ab + b²
(a - b)² = a² - 2ab + b²
a² - b² = (a - b)(a + b)
(x + a)(x + b) = x² + (a+b)x + ab

4. Division of Algebraic Expressions

Monomial by Monomial: Divide the numerical coefficients and use laws of exponents for the variables.
Polynomial by Monomial: Divide each term of the polynomial by the monomial.
Polynomial by Polynomial: Factorise the numerator polynomial and cancel out the common factors with the denominator polynomial.

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