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)

8th Grade Math Practice

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