# What Are The Types Of Factorization

## 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}}+2+frac}},xne 0,yne 0)

Solution:

Example 5:    Factorize of the following expression

(+4left( 5x-frac{1}{x} right)+4,xne 0)

Solution:

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:

Example 7:    Factorize 4x2 + 12 xy + 9 y2

Solution:

Example 8:    Factorize each of the following expressions

(i) 9x2 – 4y2

(ii) x3 – x

Solution:

Example 9:    Factorize each of the following expressions

(i) 36x2 – 12x + 1 – 25y2

(text{(ii) }-frac{9}},ane 0)

Solution:

Example 10:    Factorize the following algebraic expression

x4 – 81y4

Solution:

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:

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