# Factorization by Regrouping

In factorization by regrouping sometimes the terms of the given 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.

Method of factoring terms:

Step 1: Arrange the terms of the given expression in groups in such a way that all the groups have a common factor.

Step 2: Factorize each group.

Step 3: Take out the factor which is common to each group.

Solved problems on factorization by regrouping the terms:

1. How to factorize the following expressions?

(i) a2 + bc + ab + ac

Solution:

The expression is a2 + bc + ab + ac

By suitably rearranging the terms, we have;

= a2 + ab + ac + bc

= a(a + b) + c(a + b)

= (a + b) (a + c).

(ii) ax2 + by2 + bx2 + ay2

Solution:

The expression is ax2 + by2 + bx2 + ay2

By suitably rearranging the terms, we have;

= ax2 + ay2 + bx2 + by2

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

= (x2 + y2) (a + b).

2. Factor grouping the algebraic expressions:

(i) xy - pq + qy - px

Solution:

xy - pq + qy - px

By suitably rearranging the terms, we have;

= (xy - px) + (qy - pq)

= x (y - p) + q (y - p)

= (y - p) (x + q).

Therefore by factoring expressions we get (y - p) (x + q).

(ii) 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) + (aby2 + bxy)

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

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

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

Therefore by factoring expressions we get (bx + ay) (ax + by)