4.3 Multiplying Two Binomials

Leanne Thompson

4.3 Multiplying Two Binomials

To multiply a binomial by a binomial, we can use the Distributive Property. The two terms of the first binomial need to be multiplied by the two terms of the second binomial.

Table 4.3.1
Distributive Property for Binomials
If a, b, c, and d are real numbers, then $(a+b)(c+d)=a(c+d)+b(c+d)=ac+ad+bc+bd$

Example 1

Multiply: $\left(y+5\right)\left(y+8\right)$

Table 4.3.2
Steps	Solution
Multiply both terms by y in the first binomial. Multiply both terms by 5 in the second binomial.	$y(y+8)+5(y+8)$
Distribute.	$y^2+8y+5y+40$
Combine like terms.	$y^2+13y+40$

To help you memorize that $(a+b)(c+d) = ac+ad+bc+bd$ , you can use the acronym FOIL:

F – first terms (that is, ac)
O – outside terms (that is, ad)
I – inside terms (that is, bc)
L – last terms (that is, bd)

Practice 1

Multiply.

a) $\left(x+8\right)\left(x+9\right)$

b) $\left(2y+5\right)\left(3y+4\right)$

c) $\left(4y+3\right)\left(2y-5\right)$

d) $\left(5y+2\right)\left(6y-3\right)$

e) $\left(x+2\right)\left(x-y\right)$

f) $\left(3n+4\right)\left(n-1\right)$

Practice 2

Determine each product.

a) $\left( 3b^2 + 5 \right)\left( 4b + 6 \right)$

b) $\left( 2x^2 + 3 \right)\left( 5x - 2 \right)$

c) $\left( 7a^2 + 2 \right)\left( 4a + 3 \right)$

d) $\left( 4x - 3 \right)\left( 5x^2 - 1 \right)$

e) $\left( 3y^2 + 4 \right)\left( 2y - 1 \right)$

f) $\left( 5p - 2 \right)\left( 3p^2 - 1 \right)$

Practice 3

A square has side lengths of 3p – 4 ft. Write an expression to represent the area of the square. Draw a diagram.

Practice 4

A rectangle has a length of 3n²+ 4 km and a width of 2n – 6 km. Write an expression to represent the area of the rectangle. Draw a diagram.

Homework

Multiply.

a) $\left(5x+9\right)\left(4x+3\right)$

b) $\left(3b+5\right)\left(4b+6\right)$

c) $\left(a+10\right)\left(a+7\right)$

d) $\left(3c+4\right)\left(5c-2\right)$

e) $\left(a+7\right)\left(a-b\right)$

f) $\left(x+5\right)\left(x-y\right)$

g) $\left(x+6\right)\left(x+8\right)$

h) $\left(y+17\right)\left(y+3\right)$
Determine each product.

a) $\left(x-7\right)\left(x+5\right)$

b) $\left(b-3\right)\left(b+6\right)$

c) $\left(3x+7\right)\left(5x-2\right)$

d) $\left(4y+5\right)\left(4y-10\right)$
Expand and simplify where possible.

a) $\left(10c-d\right)\left(c-6\right)$

b) $\left(7x-y\right)\left(2x-5\right)$

c) $\left({x}^{2}+6\right)\left(x-8\right)$

d) $\left({y}^{2}+7\right)\left(y-9\right)$
Expand and simplify where possible.

a) $\left( 6b - 1 \right)\left( 4b^2 + 2 \right)$

b) $\left( 2a^2 + 5 \right)\left( a - 2 \right)$

c) $\left( 3x - 1 \right)\left( 4x^2 + 2 \right)$

d) $\left( 6z + 4 \right)\left( 2z^2 - 3 \right)$

e) $\left( 5m^2 - 7 \right)\left( 3m + 2 \right)$

f) $\left( 7y + 1 \right)\left( 2y^2 - 4 \right)$
A rectangle has a length of 3x – 2 in and a width of 2x + 5 in. Write an expression to represent the area of the rectangle. Draw a diagram.
A square has side lengths of 8c + 5 m. Write an expression to represent the area of the rectangle. Draw a diagram.

Answers

1.

a) $20{x}^{2}+51x+27$	b) $12{b}^{2}+38b+30$
c) ${a}^{2}+17a+70$	d) $15{c}^{2}+14c-8$
e) ${a}^{2}-ab+7a-7b$	f) ${x}^{2}-xy+5x-5y$
g) ${x}^{2}+14x+48$	h) ${y}^{2}+20y+51$

2.

a) ${x}^{2}-2x-35$	b) ${b}^{2}+3b-18$
c) $15{x}^{2}+29x-14$	d) $16{y}^{2}-20y-50$

3.

a) $10{c}^{2}-60c-cd+6d$	b) $14{x}^{2}-35x-2xy+5y$
c) ${x}^{3}-8{x}^{2}+6x-48$	d) ${y}^{3}-9{y}^{2}+7y-63$

4.

a) $24b^3 + 12b - 4b^2 - 2$	b) $2a^3 - 4a^2 + 5a - 10$
c) $12x^3 + 6x - 4x^2 - 2$	d) $12z^3 - 18z + 8z^2 - 12$
e) $15m^3 + 10m^2 - 21m - 14$	f) $14y^3 - 28y + 2y^2 - 4$