MULTIPLYING BINOMIALS

In this section, you will learn how to multiply binomials. 

Every term in first binomial has to be multiplied by every term in the second binomial.

Let us look at some examples to understand the above concept. 

Example 1 :

Simplify :

(2a + 3b)(5a + 4b)

Answer :

=  (2a + 3b)(5a + 4b)

=  2a(5a) + 2a(4b) + 3b(5a) + 3b(4b)

=  10a² + 8ab + 15ab + 12b²

Combine the like terms.

=  10a² + 23ab + 12b²

Example 2 :

Simplify :

(5 - 2y)(4 + y)

Answer :

=  (5 - 2y)(4 + y)

=  5(4) + 5(y) - 2y(4) - 2y(y)

=  20 + 5y - 8y - 2y²

Combine the like terms.

=  20 - 3y - 2y²

Example 3 :

Simplify : 

(3 + 2p)(4p - 3)

Answer :

=  (3 + 2p)(4p - 3)

=  3(4p) + 3(-3) + 2p(4p) + 2p(-3)

=  12p - 9 + 8p² - 6p

Combine the like terms.

=  8p² + 6p - 9

Example 4 :

Simplify :

(2m + 3)(m + 5)

Answer :

=  (2m + 3)(m + 5)

=  2m(m) + 2m(5) + 3(m) + 3(5)

=  2m² + 10m + 3m + 15

Combine the like terms. 

=  2m² + 13m + 15

Example 5 :

Simplify :

(3p - 2q)(p + 5q)

Answer :

=  (3p - 2q)(p + 5q)

=  3p(p) + 3p(5q) - 2q(p) - 2q(5q)

=  3p² + 15pq - 2pq - 10q²

Combining the like terms

=  3p² + 13pq - 10q²

Example 6 :

Simplify :

(5a + 3b)(a - 2b)

Answer :

=  (5a + 3b)(a - 2b)

=  5a(a) + 5a(-2b) + 3b(a) + 3b(-2b)

=  5a² - 10ab + 3ab - 6b²

Combine the like terms.

=  5a² - 7ab - 6b²

Example 7 :

Simplify : 

(p - 2)(p - 3)

Answer :

=  (p - 2)(p - 3)

=  p(p) + p(-3) - 2(p) - 2(-3)

=  p² - 3p - 2p + 6

Combine the like terms.

=  p² - 5p + 6

Example 8 :

Simplify : 

(p² - 3)(p + 4)

Answer :

=  (p² - 3)(p + 4)

=  p²(p) + p²(4) - 3(p) - 3(4)

=  p³ + 4p² - 3p - 12

Example 9 :

Simplify : 

(t + 5)(t - 5)

Answer :

=  (t + 5)(t - 5)

Using the identity, a² - b²  =  (a + b)(a - b),

=  t² - 5²

=  t² - 25

Example 10 :

Simplify : 

(2p + 3q)(2p - 3q)

Answer :

=  (2p + 3q)(2p - 3q)

Using the identity, a² - b²  =  (a + b)(a - b),

=  (2p)² - (3q)²

=  2²p² - 3²q²

=  4p² - 9q²

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