Factorization of Polynomials
Form 2
There are four methods of factorization:
1. Extracting common factors
2. Grouping
3. Using identities
4. Cross method
1. Extracting common factors
In this method, we rewrite the expression with the common factors outside the
brackets.
a. 2x – 8
= 2(x – 4)
b. 2x(3x – 1) – 7(3x – 1)
= (3x – 1) (2x – 7)
c. 5x(2x + 3) – 2x – 3
= 5x(2x + 3) – 1(2x + 3)
= (2x +3) (5x – 1)
2. Grouping
Expressions containing 4 or more terms can be factorized by the method of
grouping. In this method, the terms are divided into groups that the terms in
each group have a common factor.
a. 5x + 5y + ax + ay
= 5(x + y) + a(x + y)
= (x + y) (5 + a)
b. 15ax – 2by – 3bx + 10ay
= 15ax – 3bx + 10ay – 2by
= 3x(5a – b) + 2y(5a – b)
= (5a – b) (3x +2y)
3. Using identities
Perfect squares:
i. (a + b)2 ≡ a2 + 2ab + b2
ii. (a – b)2 ≡ a2 – 2ab + b2
Different of two squares:
iii. a2 – b2 ≡ (a – b) (a + b)
a. x2 + 2x + 1
= x2 + 2(x)(1) + 12
= (x + 1)2
b. 9x2 – 12xy + 4y2
= (3x)2 – 2(3x)(2y) + (2y)2
= (3x – 2y)2
c. 81x4 – y4
= (9x2)2 – (y2)2
= (9x2 – y2) (9x2 + y2)
= [(3x)2 – y2] (9x2 + y2)
= (3x – y) (3x + y) (9x2 + y2)
4. Cross method
a. x2 – 5x + 6 c. 8x2z – 16xyz + 6y2z
= (x – 2) (x – 3) = 2z (4x2 – 8xy + 3y2)
= 2z (2x – y) (2x – 3y)
x2 +6
x –2 4x2 3y2
x –3 2x –y
–3x – 2x = –5x 2x –3y
–6xy – 2xy = –8xy
b. 1 – 2x – 15x2
= –15x2 – 2x + 1
= (3x +1) (-5x + 1)
–15x2 +1
3x 1
–5x 1
3x – 5x = –2x
