# Factorize by Regrouping The Terms

In factorize by regrouping the terms sometimes it is observed that all the terms of the expression do not have any common factor, neither a monomial nor a binomial.

Follow the steps to factorize by regrouping the terms:

Step 1: From the algebraic expression arrange the groups of the given expression in such a way, that a common factor can be taken out from each group.

Step 2: Factorize each group.

Step 3: Now take out the common factor of the groups formed.

Examples to factorize algebraic expressions:

1. Factoring the following expressions

(i) ab (x2 + y2) - xy (a2 + b2)

Solution:

ab(x2 + y2) - xy(a2 + b2)

By suitably rearranging the terms, we have;

= abx2 + aby2 - a2xy - b2xy

= abx2 - a2xy - b2xy + aby2

= ax(bx - ay) - by(bx - ay)

= (bx - ay) (ax - by)

(ii) 2ax – 4ay - 3bx + 6y

Solution:

2ax – 4ay - 3bx + 6y

By suitably rearranging the terms, we have;

= 2ax – 3bx – 4ay + 6by

= x(2a – 3b) - 2y(2a – 3b)

= (2a – 3b) (x - 2y)

(iii) - 5 - 10t + 20t2

Solution:

- 5 - 10t + 20t2

By suitably rearranging the terms, we have;

= 20t2 - 10t - 5

= 5(4t2 - 2t - 1)

2. Factorize the expression:

(i) ab – a – b + 1

Solution:

ab – a – b + 1

By suitably rearranging the terms, we have;

= ab – b – a + 1

= b(a - 1) - 1(a - 1)

= (a - 1) (b - 1)

(ii) ax + ay - bx – by

Solution:

ax + ay - bx – by

By suitably rearranging the terms, we have;

= ax - bx + ay - by

= (ax - bx) + (ay - by)

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

= (a - b) (x + y)