# Binomial is a Common Factor

Factorization of algebraic expressions when a binomial is a common factor:

The expression is written as the product of binomial and the quotient obtained by dividing the given expression is by its binomial.

Solved examples when a binomial is a common factor:

1. Factorize the expression (3x + 1)2 – 5(3x + 1)

Solution:

(3x + 1)2 – 5(3x + 1)

The two terms in the above expression are (3x + 1)2 and 5(3x + 1)

= (3x + 1) (3x + 1) – 5(3x + 1)

Here, we observe that the binomial (3x + 1) is common to both the terms.

= (3x + 1) [(3x + 1) – 5]; [taking common (3x + 1)]

= (3x + 1) (3x - 4)

Therefore, (3x + 1) and (3x - 4) are two factors of the given algebraic expression.

2. Factorize the algebraic expression 2a(b - c) + 3(b – c)

Solution:

2a(b - c) + 3(b – c)

The two terms in the above expression are 2a(b - c), 3(b – c)

Here, we observe that the binomial (b – c) is common to both the terms, then we get

= 2a(b – c) + 3(b – c)

= (b – c) [2a + 3]; [taking common (b – c)]

Therefore, (b – c) and (2a + 3) are two factors of the given algebraic expression.

3. Factorize the expression (2a – 3b) (x – y) + (3a – 2b) (x – y)

Solution:

(2a – 3b) (x – y) + (3a – 2b) (x – y)

The two terms in the above expression are (2a – 3b) (x – y) and (3a – 2b) (x – y)

Here, we observe that the binomial (x – y) is common to both the terms, then we get

= (x – y) [(2a – 3b) + (3a – 2b)]

= (x – y) [(2a – 3b) + (3a – 2b)]

= (x – y) [2a – 3b + 3a – 2b]

= (x – y) [5a - 5b]

Taking common 5, we get

= (x – y) 5(a – b)

= 5(x – y) (a – b)

Therefore, 5, (x – y) and (a – b) are three factors of the given algebraic expression.

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