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:

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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:

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: