MP Board Class 8th Maths Solutions Chapter 14 Factorization Ex 14.2
MP Board Class 8th Maths Solutions Chapter 14 Factorization Ex 14.2
MP Board Class 8th Maths Solutions Chapter 14 Factorization Ex 14.2
Question 1.
Factorise the following expressions.
(i) a2 + 8o + 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 (Hint: Expand (l + m)2 first)
(viii) a4 + 2a2b2 + b4
Solution:
(i) The expression is a2 + 8a +16
= a2 + 2 × 4 × a + (4)2
= (a + 4)2 = (a + 4) (a + 4)
(ii) The expression is p2 – 10p + 25
= (p)2 – 2 × 5 × p + (5)2
= (P – 5)2 = (p – 5)(p – 5)
(iii) The expression is 25m2 + 30m + 9
= (5m)2 + 2 × 3 × (5m) + (3)2 = (5m + 3)2
= (5m + 3) (5m + 3)
(iv) The expression is 49y2 + 84yz + 36z2
= (7y)2 + 2 × (7y) × (6z) + (6z)2 = (7y + 6z)2
= (7y + 6z)(7y + 6z)
(v) The expression is 4x2 – 8x + 4
= (2x)2 – 2 × (2x) × 2 + (2)2 = (2x – 2)2
= [2(x – 1)]2 = 4(x – 1)2
= 4(x – 1)(x – 1)
(vi) The expression is 121b2 – 88bc + 16c2
= (11b)2 – 2 × (11b) (4c) + (4c)2 = (11b – 4c)2
= (11b – 4c)(11b – 4c)
(vii) The expression is (l + m)2 – 4lm
= l2 + 2 × l × m + m2 – 4lm [ ∵ (a + b)2 = a2 + 2ab + b2]
= l2 + 2lm + m2 – 4lm = l2 – 2lm + m2
= (l – m)2 = (l – m)(l – m)
(viii)The expression is a4 + 2a2b2 + b4
= (a2)2 + 2 × a2 × b2 + (b2)2 = (a2 + b2)2
= (a2 + b2)(a2 + b2)
Question 2.
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.
Solution:
(i) The expression is 4p2 – 9q2
= (2P)2 – (3q)2 = (2p + 3q) (2p – 3q)
(ii) The expression is 63a2 – 112b2
= 7[9a2 – 16b2] = 7[(30)2 – (4b)2]
= 7(3o + 4b) (3a – 4b).
(iii) The expression is 49x2 – 36 = (7x)2 – (6)2
= (7x + 6) (7x – 6).
(iv) The expression is 16x5 – 144x3
= 16x3(x2 – 9) = 16x3 (x2 – 32)
= 16x3 (x + 3)(x – 3).
(v) The expression is (l + m)2 – (l – m)2
= (l2 + 2lm + m2) – (l2 – 2lm + m2)
= l2 + 2lm + m2 – l2 + 2lm – m2 = 4lm
(vi) The expression is 9x2y2 – 16
= (3xy)2 – (4)2 = (3xy + 4) (3xy – 4)
(vii) The expression is (x2 – 2xy + y2) – z2
= (x – y)2 – z2 = (x – y + z) (x – y – z)
(viii)The expression is 25a2 – 4b2 + 28bc – 49c2
= 25a2 – [4b2 – 28bc + 49c2]
= 25a2 – [(2b)2 – 2 × (2b) × (7c) + (7c)2]
= (5a)2 – (2b – 7c)2
= (5a + 2b – 7c) (5a – 2b + 7c)
Question 3.
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
Solution:
(i) The expression is ax2 + bx = x(ax + b)
(ii) The expression is 7p2 + 21q2 = 7(p2 + 3q2 )
(iii) The expression is 2x3 + 2xy2 + 2xz2
= 2x(x2 + y2 + z2 ).
(iv) The expression is am2 + bm2 + bn2 + an2
= m2 (a + b) + n2 (b + a) = (m2 + n2 ) (a + b)
(v) The expression is (lm + l) + m + 1
= l(m + 1) + 1 (m + 1)= (l + 1) (m + 1)
(vi) The expression is y(y + z) + 9(y + z)
= (y + 9) (y + z).
(vii) The expression is 5y2 – 20y – 8z + 2yz
= 5y(y – 4) + 2z(y – 4) = (5y + 2z)(y – 4)
(viii) The expression is 10ab + 4a + 5b + 2
= 2a(5b + 2) + 1 (5b + 2) = (2a + 1) (5b + 2)
(ix) The expression is 6xy – 4y + 6 – 9x
= 2y(3x – 2) – 3 (3x – 2) = (2y – 3) (3x – 2)
Question 4.
Factorise.
(i) a4 – b4
(ii) p4 – 81
(iii) x4 – (y + z)4
(iv) x4 – (x – z)4
(v) a4 – 2a2b2 + b4
Solution:
(i) The expression is a4 – b2
= (a2)2 – (b2)2
= (a2 + b2) (a2 – b2)
= (a2 + b2)(a + b)(a – b).
(ii) The expression is p4 – 81 = (p)4 – (3)4
= (P2)2 – (32)2 = (p2 + 32) (p2 – 32)
= (p2 + 9) (p + 3) (p – 3)
(iii) The expression is x4 – (y + z)4
= (x2)2 – ((y + z)2)2
= [x2 + (y + z)2] [x2 – (y + z)2]
= [x2 + (y + z)2] (x + y + z) (x – (y + z))
= [x2 + (y + z)2] (x + y + z) (x – y – z)
(iv) The expression is x4 – (x – z)4
= (x2)2 – ((x – z)2)2
= (x2 – (x – z)2)(x2 + (x – z)2)
= (x – x + z)(x + x – z)(x2 + x2 + z2 – 2xz)
= z(2x – z) (2x2 – 2xz + z2).
(v) The expression is a4 – 2a2b2 + b4
= (a2)2 – 2(a2) (b2) + (b2)2 = (a2 – b2)2
= [(a + b) (a – b)]2 = (a + b)2 (a – b)2
Question 5.
Factorise the following expressions.
(i) p2 + 6p + 8
(ii) q2 – 10q + 21
(iii) p2 + 6p – 16
Solution:
(i) The expression is p2 + 6p + 8
= p2 + 4p + 2p + 8 = p(p + 4) + 2 (p + 4)
= (p + 2) (p + 4)
(ii) The expression is q2 – 10q + 21
= q2 – 7q – 3q + 21 = q(q – 7) – 3 (q – 7)
= (q – 3) (q – 7)
(iii) The expression is p2 + 6p – 16
= p2 + 8p – 2p – 16
= p(p + 8) – 2(p + 8) = (p – 2) (p + 8)
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