What Are The Types Of Factorization

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Types Of Factorization Example Problems With Solutions

Type I: Factorization by taking out the common factors. 

Example 1:    Factorize the following expression
2x2y + 6xy2 + 10x2y2
Solution:    2x2y + 6xy2 + 10x2y2
=2xy(x + 3y + 5xy)

Type II: Factorization by grouping the terms.   

Example 2:    Factorize the following expression
a2 – b + ab – a
Solution:    a2 – b + ab – a
= a2 + ab – b – a = (a2 + ab) – (b + a)
= a (a + b) – (a + b) = (a + b) (a – 1)

Type III: Factorization by making a perfect square.  

Example 3:    Factorize of the following expression
9x2 + 12xy + 4y2
Solution:    9x2 + 12xy + 4y2
= (3x)2 + 2 × (3x) × (2y) + (2y)2
= (3x + 2y)2

Example 4:    Factorize of the following expression
\(\frac{{{x}^{2}}}{{{y}^{2}}}+2+\frac{{{y}^{2}}}{{{x}^{2\prime }}},x\ne 0,y\ne 0\)
Solution:    
Factorization-by-perfect-square-Example-1

Example 5:    Factorize of the following expression
\({{\left( 5x-\frac{1}{x} \right)}^{2}}+4\left( 5x-\frac{1}{x} \right)+4,x\ne 0\)
Solution:
Factorization-by-perfect-square-Example-2

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Type IV: Factorizing by difference of two squares.

Example 6:    Factorize the following expressions
(a) 2x2y + 6 xy2 + 10 x2y2
(b) 2x4 + 2x3y + 3xy2 + 3y3
Solution:
Factorizing-by-difference-of-two-squares-example-1

Example 7:    Factorize 4x2 + 12 xy + 9 y2
Solution:
Factorizing-by-difference-of-two-squares-example-7

Example 8:    Factorize each of the following expressions
(i) 9x2 – 4y2
(ii) x3 – x
Solution:
Factorizing-by-difference-of-two-squares-example-8

Example 9:    Factorize each of the following expressions
(i) 36x2 – 12x + 1 – 25y2
\(\text{(ii) }{{a}^{2}}-\frac{9}{{{a}^{2}}},a\ne 0\)
Solution:
Factorizing-by-difference-of-two-squares-example-9

Example 10:    Factorize the following algebraic expression
x4 – 81y4
Solution:
Factorizing-by-difference-of-two-squares-example-10

Example 11:    Factorize the following expression
x(x+z) – y (y+z)
Solution:    x(x+z) – y (y+z) = (x2 – y2) + (xz–yz)
= (x–y) (x+y) + z (x–y)
= (x–y) {(x+y) + z}
= (x–y) (x+ y + z)

Example 12:    Factorize the following expression
x4 + x2 + 1
Solution:    x4 + x2 + 1 = (x4 + 2x2 +1) – x2
= (x2 +1)2 – x2 = (x2 + 1 – x) (x2 + 1+x)
= (x2–x + 1) (x2 + x + 1)

Type V: Factorizing the sum and difference of cubes of two quantities.
(i) (a3 + b3) = (a + b) (a2 – ab + b2)
(ii) (a3 – b3) = (a – b) (a2 + ab + b2)

Example 13:    Factorize the following expression
a3 + 27
Solution:   a3  + 27 = a3  + 33 = (a + 3) (a2 –3a +9)

Example 14:    Simplify : (x+ y)3 – (x –y)3 – 6y(x2 – y2)
Solution:    
factorizing-the-sum-and-difference-of-cubes

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