# Factoring Terms by Grouping

How to factor an algebraic expression step-by-step?

Method to factoring algebraic expression by grouping:

(i) From the groups of the given expression a factor can be taken out from each group.

(ii) Factorize each group

(iii) Now take out the factor common to group formed.

Now we will learn how to factor the terms by grouping.

Solved examples of factoring terms by grouping:

1. Factoring of algebraic expression:

(i) 2ax + ay + 2bx + by

Solution:

2ax + ay + 2bx + by

= a(2x + y) + b(2x + y)

= (2x + y) (a + b)

(ii) 3ax - bx - 3ay + by

Solution:

3ax - bx - 3ay + by

= x(3x - b) - y(3x - b)

= (3x - b) (x - y)

(iii) 6x2 + 3xy - 2ax – ay

Solution:

6x2 + 3xy - 2ax – ay

= 3x(2x + y) - a(2x + y)

= (2x + y)(3x – a)

(iv) ax2 - bx2 + ay2 - by2 + az2 - bz2

Solution:

ax2 - bx2 + ay2 - by2 + az2 - bz2

= x2(a - b) + y2(a - b) + z2(a - b)

= (a - b)(x2 + y2 + z2)

(v) am - an + bm – bn

Solution:

am - an + bm - bn

= a(m - n) + b(m - n)

= (m - n) (a + b)

2. Factorize the following algebraic expression:

(i) 6x + 3xy + y + 2

Solution:

6x + 3xy + y + 2

= (6x + 3xy) + (y + 2)

= 3x(2 + y) + 1(2 + y)

= 3x(y + 2) + 1(y + 2)

= (y + 2) (3x + 1)

= (3x + 1) (y + 2)

(ii) 3x3 + 5x2 + 3x + 5

Solution:

3x3 + 5x2 + 3x + 5

= x2(3x + 5) + 1(3x + 5)

= (3x + 5)(x2 + 1)

(iii) x3 + 3x2 + x + 3

Solution:

x3 + 3x2 + x + 3

= (x3 + 3x2) + (x + 3)

= x2(x + 3) + 1(x + 3)

= (x + 3) (x2 + 1)

(iv) 1 + m + m2n + m3n

Solution:

1 + m + m2n + m3n

= (1 + m) + (m2n + m3n)

= 1(1 + m) + m2n(1 + m)

= (1 + m) (1 + m2n)

(v) x - 1 - (x - 1)2 + ax – a

Solution:

x - 1 - (x - 1)2 + ax – a

= 1(x - 1) - (x - 1)2 + a(x - 1)

= (x - 1) [1 - (x - 1) + a]

= (x - 1) [1 - x + 1 + a]

= (x - 1) (2 + a - x)