# Factoring Terms by Regrouping

Factoring terms by regrouping (two or more) means that you need to rearrange the terms with common factors before factoring. In case of regrouping the terms of the given algebraic expression need to be arranged in suitable groups in such a way that all the groups have a common factor. After this arrangement factorization becomes easy.

Solved examples on factoring terms by regrouping:

1. Factorize the expression:

(i) a2x + abx + ac + aby + b2y + bc

Solution:

a2x + abx + ac + aby + b2y + bc

By suitably rearranging the terms, we have;

= a2x + abx + aby + b2y + ac + bc

= ax(a + b) + by(a + b) + c(a + b)

= (a + b) (ax + by + c)

(ii) p3k + p2(k – m) – p(m + n) – n

Solution:

p3k + p2(k – m) – p (m + n) – n

By suitably rearranging the terms, we have;

= p3k + p2k – p2m – pm – pn – n

= (p3k + p2k) – (p2m + pm) – (pn + n)

= p2k (p + 1) - pm (p + 1) – n (p + 1)

= (p + 1) (p2k – pm – n)

2. How to factorize by grouping the following expressions?

(i) ax – bx + by + cy – cx – ay

Solution:

ax – bx + by + cy – cx – ay

By suitably rearranging the terms, we have;

= ax - bx – cx – ay + by + cy

= x(a – b – c) - y(a – b – c)

(a – b – c) (x - y)

(ii) x3 - 2x2 + ax + x - 2a – 2

Solution:

x3 - 2x2 + ax + x - 2a – 2

By suitably rearranging the terms, we have;

= x3 - 2x2 + ax - 2a + x - 2

= (x3 - 2x2) + (ax - 2a) + (x - 2)

= x2(x - 2) + a(x - 2) + 1(x - 2)

= (x - 2) (x2 + a + 1)