Class 8 Maths

# Factorisation

## Exercise 14.2

Question 1: Factorise the following expressions.

(i) a^2 + 8a + 16

Answer: This equation can be facorised by using the identity; (a + b)^2 = a^2 + 2ab + b^2

Factors = (a + 4)^2 = (a + 4)(a + 4)

(ii) p^2 – 10 p + 25

Answer: This equation can be factorised by using the identity; (a – b)^2 = a^2 – 2ab + b^2

Factors = (p – 5)^2

(iii) 25m^2 + 30m + 9

Answer: = (5m – 3)^2

(iv) 49y^2 + 84yz + 36z^2

Answer: (7y + 6z)^2

(v) 4x^2 – 8x + 4

Answer: (2x – 2)^2

(vi) 121b^2 – 88bc + 16c^2

Answer: (11b – 4c)^2

(vii) (l + m)^2 – 4lm

Answer: l^2 + m^2 + 2lm - 4lm
l^2+ m^2 - 2lm = (l + m)^2

(viii) a^4 + 2a^2b^2 + b^4

Answer: This can be solved using (a+b)^2 = a^2+ 2ab + b^2
Hence, a^4 + 2a^2b^2 + b^4
=(a^2+b^2)^2

Question 2: Factorise.

(i) 4p^2 – 9q^2

Answer: This can be factorised by using the equation; (a + b)(a – b) = a^2 – b^2
Factors = (2p + 3q)(2p – 3q)

(ii) 63a^2 – 112b^2

Answer: 63a^2 – 112b^2 = 7(9a^2 – 16b^2)
= 7(3a + 4b)(3a – 4b)

(iii) 49x^2 – 36

Answer: (7x + 6)(7x – 6)

(iv) 16x^5 – 144x^3

Answer: 16x^5-144x^3
= x^3(16x^2-144)
= x^3(4x+12)(4x-12)

(v) (l + m)^2 – (l – m)^2

Answer: (l + m + l – m)(l + m – l + m)
2l \xx 2m = 4lm

(vi) 9x^2 y^2 – 16

Answer: (3xy + 4)(3xy – 4)

(vii) (x^2 – 2xy + y^2) – z^2

Answer: (x^2 – 2xy + y^2) – z^2
= (x – y)^2 – z^2
= (x – y + z)(x – y – z)

(viii) 25a^2 – 4b^2 + 28bc – 49c^2

Answer: 25a^2 – 4b^2 + 28bc – 49c^2
= (5a)^2 – (2b)^2 + 2 xx 2b \xx 7c – (7c)^2
= (5a)^2 – [(2b)^2 – 2 xx 2b\ xx 7c + (7c)^2]
= (5a)^2 – (2b – 7c)^2
This can be further factorised by using (a + b)(a – b) = a^2 – b^2
= (5a + 2b – 7c)(5a – 2b + 7c)

Question 3: Factorise the expressions.

(i) ax^2 + bx

Answer: x(ax + b)

(ii) 7p^2 + 21q^2

Answer: 7(p^2 + 3q^2)

(iii) 2x^3 + 2xy^2 + 2xz^2

Answer: 2x^3 + 2xy^2 + 2xz^2
= 2x(x^2+y^2+z^2)

(iv) am^2 + bm^2 + bn^2 + an^2

Answer: a(m^2 + n^2) + b(m^2 + n^2)
= (a + b)(m^2 + n^2)

(v) (lm + l) + m + 1

Answer: l(m + 1) + 1(m + 1)
= (l + 1)(m + 1)

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

Answer: (y + 9)(y + z)

(vii) 5y^2 – 20y – 8z + 2yz

Answer: 5y(y + 4) + 2z(y + 4)
= (5 + 2z)(y + 4)

(viii) 10ab + 4a + 5b + 2

Answer: 5b + 10ab + 2 + 4a
= 5b(1 + 2a) + 2(1 + 2a)
= (5b + 2)(1 + 2a)

(ix) 6xy – 4y + 6 – 9x

Answer: 6xy – 4y + – 9x + 6
= 2y (3x – 2) - 3 (3x - 2)
= (2y – 3)(3x – 2)

Question 4: Factorise.

(i) a^4 – b^4

Answer: a^4-b^4 = (a^2+b^2)(a^2-b^2)

(ii) p^4 – 81
=(p^2+9)(p^2-9)

(iii) x^4 – (y + z)^4

Answer: x^4 – (y + z)^4
= (x^2+(y+z)^2)(x^2-(y+z)^2)
= (x^2+(y+z)^2)[(x+y+z)(x-y-z)]

(iv) x^4 – (x – z)^4

Answer: x^4 – (x – z)^4
=(x^2-(x-z)^2)(x^2+(x-z)^2)
=[(x+x-z)(x-x+z)][x^2+(x-z)^2]

(v) a^4 – 2a^2b^2 + b^4

Answer: a^4 – 2a^2b^2 + b^4
This can be factorised by using the identity; (a - b)^2 = a^2 – 2ab + b^2
Factors = (a^2 – b^2)^2 = (a^2 – b^2)(a^2 – b^2)

Question 5: Factorise the following expressions.

(i) p^2 + 6p + 8

Asnwer: p^2+6p+8
= p(p+6)+8

(ii) q^2 – 10q + 21

Answer: q^2-10q+21
= q(q-10)+21

(iii) p^2 + 6p – 16

Answer: p^2+6p-16
= p(p + 6)- 16