Miscellaneous Problems on Factorization

Miscellaneous Problems on Factorization available will give insight on the topic of factorization as well as to acquire knowledge about various types of questions asked on factorization. All the Factorization Questions available with solutions come in handy to revise the concept effectively. Answer the different problems on factorization on a regular basis to become proficient in the topic. Subject Experts have designed the different models of questions on factorization so that students can have enough practice on the same.

Do Refer:

Miscellaneous Factoring Problems with Solutions

Example 1.
Factorize the Expression 5x2+14x-3?
Solution:
Given Expression = 5x2+14x-3
=5x2+15x-x-3
=5x(x+3)-1(x+3)
=(5x-1)(x+3)

Example 2.
Factorize 8x3y – 88x2y – 224xy?
Solution:
Given Expression = 8x3y – 88x2y – 224xy
= 8xy(x2 – 11x – 26)
=8xy(x2 – 13x+2x – 26)
=8xy(x(x-13)+2(x-13))
=8xy(x+2)(x-13)

Example 3.
Factorize (a – b)3 +(b – c)3 + (c – a)3?
Solution:
Given (a – b)3 +(b – c)3 + (c – a)3
=(a – b)3 +(b – c)3 + (c – a)3(Since a+b+c=0)
=3(a-b)(b-c)(c-a)

Example 4.
If x+\(\frac{ 1}{x}\) = \(\sqrt{ 2}\), find x3+1/x3
Solution:
Given x+1/x = \(\sqrt{ 2}\)
x3+1/x3=(x+1/x)(x2-x.1/x+1/x2)
=\(\sqrt{ 2}.\)(x2+1/x2-1)
=\(\sqrt{ 2}.\)(x+1/x)2-3
=\(\sqrt{ 2}.\)((\(\sqrt{ 2}.\))2-3)
=\(\)\sqrt{ 2 *-1}
=\(\)\sqrt{ -2}

Example 5.
Find the LCM and HCF of x2 – 4x + 3 and x2 + 3x + 2?
Solution:
Factorizing given expressions we have x2 – 4x + 3 = x2-3x-x+3
=x(x-3)-1(x-3)
=(x-3)(x-1)
x2 – 3x + 2 = x2 – 2x -x+ 2
=x(x-2)-1(x-2)
=(x-2)(x-1)

By the definition of LCM, the Least Common Multiple of given expressions is (x-3)(x-1)(x-2)
By the definition of HCF, the Highest Common Factor of given expressions is (x-1)

Example 6.
Factorize the following expressions
(i)6x-42
(ii)10x2-15y2+20z2
(iii)a2+8a+16
(iv)a4-2a2b2+b2
Solution:
(i) Given 6x-42
=6(x-7)
(ii) Given 10x2-15y2+20z2
=5(2x2-3y2+4z2)
(iii) Given a2+8a+16
=a2+4a+4a+16
=a(a+4)+4(a+4)
=(a+4)(a+4)
(iv) Given a4-2a2b2+b4
 = (a2)2-2a2b2+(b2)2
=(a2-b2)2

Example 7.
Factorize  4p2 – 4p – 3?
Solution:
Given Expression =  4p2 – 4p – 3
=4p2 -6p+2p – 3
=2p(2p-3)+1(2p-3)
=(2p+1)(2p-3)

Example 8.
Find the Common Factors of the following terms
(a) 20x2y, 24xy2
(b) 63m3n2, 45mn4
Solution:
(a) 20x2y, 24xy2
20x2y = 4*5*x*x*y
24xy2=4*6*x*y*y
Common Factors for both the terms are 4xy
(b) 63m3n2, 45mn4
63m3n2 =7*9*m*m*n*n
45mn4 =9*5*m*n*n*n*n
Common Factors for both the terms are 9m2n2

Example 9.
Factorize  a3 + b3 – 3ab + 1?
Solution:
Given Expression = a3 + b3 – 3ab + 1
= a3 + b3 + 13 – 3 ∙ a ∙ b ∙ 1
= (a +b + 1)(a2 + b2 + 12 – b ∙ 1 – 1 ∙ a – ab)
= (a +b + 1)(a2 +b2 – b – a – ab + 1)

Example 10.
If a + b + c = 15, a2 + b2 + c2 = 30 and a3 + b3 + c3 = 140, find the value of abc?
Solution:
We know, a3 + b3 + c3 – 3abc = (a + b + c)(a2 + b2 + c2 – bc – ca – ab).
Therefore, 140 – 3abc = 15(30 – bc – ca – ab)…………………….. (i)
Now, (a + b + c)2 = a2 + b2 + c2 + 2bc + 2ca + 2ab
Therefore, 152 = 30 + 2(bc + ca + ab).
⟹ 2(bc + ca + ab) = 152 – 30
⟹ 2(bc + ca + ab) = 225 – 30
⟹ 2(bc + ca + ab) = 190
Therefore, bc + ca + ab = 190/2 = 95
Putting in (i), we get,
140 – 3abc = 15(30 – 95)
⟹ 140 – 3abc = 450-1425
⟹ 140-3abc = -975
⟹ 3abc = 140+975
3abc=1115
Therefore, abc ~ 371

Leave a Reply