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Also, See: Miscellaneous Problems on Factorization
Factorization Exercises with Answers
Example 1.
Factorize the expressions of the form a3+b3?
(i)27x3+729y3
(ii)1-216a3
Solution:
(i)27x3+729y3
Given Expression = 27x3+729y3
=(3x)3+(9y)3
As per the Identity a3+b3 =(a+b)(a2-ab+b2)
= (3x+9y)((3x)2-3x.9y+(9y)2)
=(3x+9y)(9x2-27xy+81y2)
(ii)1-216a3
Given Expression = 1-216a3
=(1)3-(6a)3
As per the Identity a3-b3 =(a-b)(a2+ab+b2)
= (1-6a)((1)2+1.6a+(6a)2)
=(1-6a)(1+6a+36a2)
Example 2.
Factorize 9a2b+3a2+5b+5b2a?
Solution:
Given Expression = 9a2b+3a2+5b+5b2a
= 3a2(3b+1)+5b(1+ba)
Example 3.
Factorie the Algebraic Expression x2 – 14x + 49?
Solution:
Given Algebraic Expression = x2 – 14x + 49
=x2 – 7x-7x + 49
=x(x-7)-7(x-7)
=(x-7)(x-7)
=(x-7)2
Example 4.
If a+1/a = 3, find a3+1/a3?
Solution:
Given a+1/a = 3
a3+1/a3 =?
As per the Identity a3+b3 =(a+b)(a2-ab+b2)
Simply rewriting the expression given as per the identity we have a3+1/a3 =(a+1/a )(a2-a.1/a+1/a2)
Substituting the known values we get a3+1/a3 = 3(a2+1/a2-1)
=3((a+1/a)2-3)
=3(32-3)
=3(9-3)
=3(6)
=18
Example 5.
Find the LCM and HCF of
(i)p3-9p and p2+6p+9
(ii)1 – 8x3, 1-x-2x2
Solution:
(i)p3+9p and p2+6p+9
p3+9 = p(p2-9)
=p(p-3)(p+3)
p2+6p+9 = p2+3p+3p+9
=p(p+3)+3(p+3)
=(p+3)(p+3)
By the definition of LCM, LCM of the given expression is p(p-3)(p+3)
By the definition of HCF, the HCF of the given expression is (p+3)
(ii)1 – 8x3, 2x2-x-1
1 – 8x3 = (1)3-(2x)3
=(1-2x)((1)2+1.2x+(2x)2)
=(1-2x)(1+2x+4x2)
=(1-2x)(4x2+2x+1)
2x2-x-1 = 2x2-2x+x-1
=2x(x-1)+1(x-1)
=(2x+1)(x-1)
By the definition of LCM, LCM = (1-2x)(1+2x+4x2)(2x+1)(x-1)
By the definition of HCF, HCF = 1 as it is the only common factor
Example 6.
Factorize (2x – y – z)3 + (2y – z – x)3 + (2z – x – y)3
Solution:
Given Expression = (2x – y – z)3 + (2y – z – x)3 + (2z – x – y)3
=3(2x – y – z) . (2y – z – x) (2z – x – y)[Since a3+b3+c3; a+b+c=0 = 3abc]
Example 7.
Factorize x2+ 8x + 16
Solution:
Given Expression = x2+ 8x + 16
=x2+ 4x + 4x+16
=x(x+4)+4(x+4)
=(x+4)(x+4)
Example 8.
Find the Common Factors of the following terms
(a) 24x2y, 32xy3
(b) 54m3n2, 72mn3
Solution:
(a) 24x2y, 32xy3
24x2y = 4*2*3*x*x*y
32xy3=4*4*2*x*y*y*y
Common Factors for both the terms are 8xy
(b) 54m3n2, 72mn3
54m3n2 =9*6*m*m*m*n*n
72mn3 =9*8*m*n*n*n
Common Factors for both the terms are 9mn2
Example 9.
Factorize x2 + 9x + 14?
Solution:
Given Expression = x2 + 9x + 14
=x2 + 7x +2x+ 14
=x(x+7)+2(x+7)
=(x+7)(x+2)
Example 10.
Factorize z² + 6z – 16?
Solution:
Given Expression = z² + 6z – 16
=z² + 8z-2z – 16
=z(z+8)-2(z+8)
=(z-2)(z+8)