6.2 Multiply Polynomials

Pooja Gupta

6.2 Multiply Polynomials

Learning Objectives

By the end of this section, you will be able to:

Multiply a polynomial by a monomial
Multiply a binomial by a binomial
Multiply a trinomial by a binomial

Multiply a Polynomial by a Monomial

We have used the Distributive Property to simplify expressions like $2\left(x-3\right)$ . You multiplied both terms in the parentheses, $x$ and $3$ , by 2, to get $2x-6$ . With this chapter’s new vocabulary, you can say you were multiplying a binomial, $x-3$ , by a monomial, 2

Multiplying a binomial by a monomial is nothing new for you! Here’s an example:

EXAMPLE 1

Multiply: $4\left(x+3\right)$ .

Solution


Distribute.	$4 {\color{red}\cdot} x + 4 {\color{red}\cdot} 3$
Simplify.	$4x+12$

TRY IT 1.1

Multiply: $5\left(x+7\right)$ .

Show answer

$5x+35$

TRY IT 1.2

Multiply: $3\left(y+13\right)$ .

Show answer

$3y+39$

EXAMPLE 2

Multiply: $y\left(y-2\right)$ .

Solution


Distribute.	$y {\color{red}\cdot} y - y {\color{red}\cdot} 2$
Simplify.	$y^2-2y$

TRY IT 2.1

Multiply: $x\left(x-7\right)$ .

Show answer

${x}^{2}-7x$

TRY IT 2.2

Multiply: $d\left(d-11\right)$ .

Show answer

${d}^{2}-11d$

EXAMPLE 3

Multiply: $7x\left(2x+y\right)$ .

Solution


Distribute.	$7x {\color{red}\cdot} 2x + 7x {\color{red}\cdot} y$
Simplify.	$14x^2 + 7xy$

TRY IT 3.1

Multiply: $5x\left(x+4y\right)$ .

Show answer

$5{x}^{2}+20xy$

TRY IT 3.2

Multiply: $2p\left(6p+r\right)$ .

Show answer

$12{p}^{2}+2pr$

EXAMPLE 4

Multiply: $-2y\left(4{y}^{2}+3y-5\right)$ .

Solution


Distribute.	$-2 y {\color{red} \cdot} 4 y^2+(-2 y) {\color{red} \cdot} 3 y-(-2 y) {\color{red} \cdot} 5$
Simplify.	$-8 y^3-6 y^2+10 y$

TRY IT 4.1

Multiply: $-3y\left(5{y}^{2}+8y-7\right)$ .

Show answer

$-15{y}^{3}-24{y}^{2}+21y$

TRY IT 4.2

Multiply: $4{x}^{2}\left(2{x}^{2}-3x+5\right)$ .

Show answer

$8{x}^{4}-12{x}^{3}+20{x}^{2}$

EXAMPLE 5

Multiply: $2{x}^{3}\left({x}^{2}-8x+1\right)$ .

Solution


Distribute.	$2 x^3 {\color{red} \cdot} x^2+\left(2 x^3\right) {\color{red} \cdot}(-8 x)+\left(2 x^3\right) {\color{red} \cdot} 1$
Simplify.	$2 x^5-16 x^4+2 x^3$

TRY IT 5.1

Multiply: $4x\left(3{x}^{2}-5x+3\right)$ .

Show answer

$12{x}^{3}-20{x}^{2}+12x$

TRY IT 5.2

Multiply: $-6{a}^{3}\left(3{a}^{2}-2a+6\right)$ .

Show answer

$-18{a}^{5}\ + 12{a}^{4}-36{a}^{3}$

EXAMPLE 6

Multiply: $\left(x+3\right)p$ .

Solution

The monomial is the second factor.
Distribute.	$x {\color{red} \cdot} p + 3{\color{red} \cdot} p$
Simplify.	$xp+3p$

TRY IT 6.1

Multiply: $\left(x+8\right)p$ .

Show answer

$xp+8p$

TRY IT 6.2

Multiply: $\left(a+4\right)p$ .

Show answer

$ap+4p$

Multiply a Binomial by a Binomial

Just like there are different ways to represent multiplication of numbers, there are several methods that can be used to multiply a binomial times a binomial. We will start by using the Distributive Property.

Multiply a Binomial by a Binomial Using the Distributive Property

Look at the table below, where we multiplied a binomial by a monomial.

x plus 3, in parentheses, times p. Two arrows extend from the p, terminating at x and 3.

We distributed the p to get:

What if we have (x + 7) instead of p?

x plus 3 multiplied by x plus 7. Two arrows extend from x plus 7, terminating at the x and the 3 in the first binomial.

Distribute (x + 7).

The sum of two products. The product of x and x plus 7, plus the product of 3 and x plus 7.

Distribute again.

Combine like terms.

Notice that before combining like terms, you had four terms. You multiplied the two terms of the first binomial by the two terms of the second binomial—four multiplications.

EXAMPLE 7

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

Solution


Distribute (y + 8).	$y {\color{red}{(y+8)}}+5{\color{red}{(y+8)}}$
Distribute again	$y {\color{red}{\cdot}} y +y {\color{red}{\cdot}} 8 + 5 {\color{red}{\cdot}} y + 5 {\color{red}{\cdot}} 8$ $y^2+8 y+5 y+40$
Combine like terms.	$y^2+13 y+40$

TRY IT 7.1

Multiply: $\left(x+8\right)\left(x+9\right)$ .

Show answer

${x}^{2}+17x+72$

TRY IT 7.2

Multiply: $\left(5x+9\right)\left(4x+3\right)$ .

Show answer

$20{x}^{2}+51x+27$

EXAMPLE 8

Multiply: $\left(2y+5\right)\left(3y+4\right)$ .

Solution


Distribute (3y + 4).	$2 y {\color{red}(3 y+4)}+5 {\color{red}(3 y+4)}$
Distribute again	$2y {\color{red}{\cdot}} 3y + 2y {\color{red}{\cdot}} 4 + 5 {\color{red}{\cdot}} 3y + 5 {\color{red}{\cdot}} 4$ $6 y^2+8 y+15 y+20$
Combine like terms.	$6 y^2+23 y+20$

TRY IT 8.1

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

Show answer

$12{b}^{2}+38b+30$

TRY IT 8.2

Multiply: $\left(a+10\right)\left(a+7\right)$ .

Show answer

${a}^{2}+17a+70$

EXAMPLE 9

Multiply: $\left(4y+3\right)\left(2y-5\right)$ .

Solution


Distribute.	$4 y {\color{red}{(2 y-5)}}+3{\color{red}{(2 y-5)}}$
Distribute again.	$4y {\color{red}{\cdot}} 2y + 4y {\color{red}{\cdot}} (-5) + 3 {\color{red}{\cdot}} 2y + 3 {\color{red}{\cdot}} (-5)$ $8 y^2-20 y+6 y-15$
Combine like terms.	$8 y^2-14 y-15$

TRY IT 9.1

Multiply: $\left(5y+2\right)\left(6y-3\right)$ .

Show answer

$30{y}^{2}-3y-6$

TRY IT 9.2

Multiply: $\left(3c+4\right)\left(5c-2\right)$ .

Show answer

$15{c}^{2}+14c-8$

EXAMPLE 10

Multiply: $\left(x+2\right)\left(x-y\right)$ .

Solution


Distribute.	$x {\color{red}{(x-y)}}-2{\color{red}{(x-y)}}$
Distribute again.	$x {\color{red}{\cdot}} (x) + x {\color{red}{\cdot}} (-y) + (-2) {\color{red}{\cdot}} (x) + (-2) {\color{red}{\cdot}} (-y)$ $x^2-x y-2 x+2 y$
There are no like terms to combine.	$x^2-x y-2 x+2 y$

TRY IT 10.1

Multiply: $\left(a+7\right)\left(a-b\right)$ .

Show answer

${a}^{2}-ab+7a-7b$

TRY IT 10.2

Multiply: $\left(x+5\right)\left(x-y\right)$ .

Show answer

${x}^{2}-xy+5x-5y$

Multiply a Binomial by a Binomial Using the FOIL Method

Remember that when you multiply a binomial by a binomial you get four terms. Sometimes you can combine like terms to get a trinomial, but sometimes, like in the above example, there are no like terms to combine.

Let’s look at the last example again and pay particular attention to how we got the four terms.

$\begin{array}{c}\hfill \left(x-2\right)\left(x-y\right)\hfill \\ \hfill {x}^{2}-xy-2x+2y\hfill \end{array}$

Where did the first term, ${x}^{2}$ , come from?

We abbreviate “First, Outer, Inner, Last” as FOIL. The letters stand for ‘First, Outer, Inner, Last’. The word FOIL is easy to remember and ensures we find all four products.
FOIL image

$(x-2)(x-y)$

$\begin{matrix}{x}^{2}- xy- 2x+ 2y \\ F \quad \enspace O \quad \enspace I \quad \enspace L \end{matrix}$

Let’s look at $\left(x+3\right)\left(x+7\right)$ .

Distibutive Property

FOIL

The product of x plus 3 and x plus y. An arrow extends from the x in x plus 3 to the x in x plus 7. A second arrow extends from the x in x plus 3 to the 7 in x plus 7. A third arrow extends from the 3 in x plus 3 to the x in x plus 7. A fourth arrow extends from the 3 in x plus 3 to the 7 in x plus 7.

The sum of two products, the product of x and x plus 7, and the product of 3 and x plus 7.

x squared plus 7 x plus 3 x plus 21. Below x squared is the letter F, below 7 x is the letter O, below 3 x is the letter I, and below 21 is the letter L, spelling FOIL.

Notice how the terms in third line fit the FOIL pattern.

Now we will do an example where we use the FOIL pattern to multiply two binomials.

EXAMPLE 11

How to Multiply a Binomial by a Binomial using the FOIL Method

Multiply using the FOIL method: $\left(x+5\right)\left(x+9\right)$ .

Solution

$\left(x+5\right)\left(x+9\right)$
Step 1. Multiply the First terms, $\color{red} x \cdot x$		$\begin{array}{ccccccc} {\color{red}{x^2}} & + & \rule{.5cm}{0.4pt} & + & \rule{.5cm}{0.4pt} & + & \rule{.5cm}{0.4pt} \\ F & & O & & I & & L \\ \end{array}$
Step 2. Multiply the Outer terms, $\color{red} x \cdot 9$		$\begin{array}{ccccccc} x^2 & {\color{red}{+}} &{\color{red}{9x}} & + & \rule{.5cm}{0.4pt} & + & \rule{.5cm}{0.4pt} \\ F & & O & & I & & L \\ \end{array}$
Step 3. Multiply the Inner terms, $\color{red} 5 \cdot x$		$\begin{array}{ccccccc} x^2 & + & 9x & + & {\color{red}{5x}} & + & \rule{.5cm}{0.4pt} \\ F & & O & & I & & L \\ \end{array}$
Step 4. Multiply the Last terms, $\color{red} 5 \cdot 9$		$\begin{array}{ccccccc} x^2 & + & 9x & + & 5x & {\color{red}{+}}& {\color{red}{45}} \\ F & & O & & I & & L \\ \end{array}$
Step 5. Combine like terms.		$x^2+14x+45$

TRY IT 11.1

Multiply using the FOIL method: $\left(x+6\right)\left(x+8\right)$ .

Show answer

${x}^{2}+14x+48$

TRY IT 11.2

Multiply using the FOIL method: $\left(y+17\right)\left(y+3\right)$ .

Show answer

${y}^{2}+20y+51$

We summarize the steps of the FOIL method below. The FOIL method only applies to multiplying binomials, not other polynomials!

HOW TO: Multiply two binomials using the FOIL method

When you multiply by the FOIL method, drawing the lines will help your brain focus on the pattern and make it easier to apply.

EXAMPLE 12

Multiply: $\left(y-7\right)\left(y+4\right)$ .

Solution

$\left(y-7\right)\left(y+4\right)$
Multiply the First terms, $\color{red} y \cdot y$		$\begin{array}{ccccccc} {\color{red}{y^2}} & + & \rule{.5cm}{0.4pt} & + & \rule{.5cm}{0.4pt} & + & \rule{.5cm}{0.4pt} \\ F & & O & & I & & L \\ \end{array}$
Multiply the Outer terms, $\color{red} y \cdot 4$		$\begin{array}{ccccccc} y^2 & {\color{red}{+}} &{\color{red}{4y}} & + & \rule{.5cm}{0.4pt} & + & \rule{.5cm}{0.4pt} \\ F & & O & & I & & L \\ \end{array}$
Multiply the Inner terms, $\color{red} -7 \cdot y$		$\begin{array}{ccccccc} y^2 & + & 4y & {\color{red}{-}} & {\color{red}{7y}} & + & \rule{.5cm}{0.4pt} \\ F & & O & & I & & L \\ \end{array}$
Multiply the Last terms, $\color{red} -7 \cdot 4$		$\begin{array}{ccccccc} y^2 & + & 4y & - & 7y & {\color{red}{-}}& {\color{red}{28}} \\ F & & O & & I & & L \\ \end{array}$
Combine like terms.		$y^2-3y-28$

TRY IT 12.1

Multiply: $\left(x-7\right)\left(x+5\right)$ .

Show answer

${x}^{2}-2x-35$

TRY IT 12.2

Multiply: $\left(b-3\right)\left(b+6\right)$ .

Show answer

${b}^{2}+3b-18$

EXAMPLE 13

Multiply: $\left(4x+3\right)\left(2x-5\right)$ .

Solution

$\left(4x+3\right)\left(2x-5\right)$
Multiply the First terms, $\color{red} 4x \cdot 2x$	$\begin{array}{ccccccc} {\color{red}{8x^2}} & + & \rule{.5cm}{0.4pt} & + & \rule{.5cm}{0.4pt} & + & \rule{.5cm}{0.4pt} \\ F & & O & & I & & L \\ \end{array}$
Multiply the Outer terms, $\color{red} 4x \cdot (-5)$	$\begin{array}{ccccccc} 8x^2 & {\color{red}{-}} &{\color{red}{20x}} & + & \rule{.5cm}{0.4pt} & + & \rule{.5cm}{0.4pt} \\ F & & O & & I & & L \\ \end{array}$
Multiply the Inner terms, $\color{red} 3 \cdot 2x$	$\begin{array}{ccccccc} 8x^2 & - & 20x & {\color{red}{+}} & {\color{red}{6x}} & + & \rule{.5cm}{0.4pt} \\ F & & O & & I & & L \\ \end{array}$
Multiply the Last terms, $\color{red} 3 \cdot (-5)$	$\begin{array}{ccccccc} 8x^2 & - &20x & + & 6x & {\color{red}{-}}& {\color{red}{15}} \\ F & & O & & I & & L \\ \end{array}$
Combine like terms.	$8x^2-14x-15$

TRY IT 13.1

Multiply: $\left(3x+7\right)\left(5x-2\right)$ .

Show answer

$15{x}^{2}+29x-14$

TRY IT 13.2

Multiply: $\left(4y+5\right)\left(4y-10\right)$ .

Show answer

$16{y}^{2}-20y-50$

The final products in the last four examples were trinomials because we could combine the two middle terms. This is not always the case.

EXAMPLE 14

Multiply: $\left(3x-y\right)\left(2x-5\right)$ .

Solution

$\left(3x-y\right)\left(2x-5\right)$
Multiply the First.	$\begin{array}{ccccccc} {\color{red}{6x^2}} & + & \rule{0.5cm}{0.4pt} & + & \rule{0.5cm}{0.4pt} &+& \rule{0.5cm}{0.4pt} \\ F&&O&&I&&L \end{array}$
Multiply the Outer.	$\begin{array}{ccccccc} 6x^2 & {\color{red}{-}} & {\color{red}{15x}} & + & \rule{0.5cm}{0.4pt} &+& \rule{0.5cm}{0.4pt} \\ F&&O&&I&&L \end{array}$
Multiply the Inner.	$\begin{array}{ccccccc} 6x^2 & - & 15x & {\color{red}{-}} & {\color{red}{2xy}} &+& \rule{0.5cm}{0.4pt} \\ F&&O&&I&&L \end{array}$
Multiply the Last.	$\begin{array}{ccccccc} 6x^2 & - & 15x & - & 2xy & {\color{red}{+}}& {\color{red}{5y}} \\ F&&O&&I&&L \end{array}$
Combine like terms—there are none.	$6 x^2-15 x-2 x y+5 y$

TRY IT 14.1

Multiply: $\left(10c-d\right)\left(c-6\right)$ .

Show answer

$10{c}^{2}-60c-cd+6d$

TRY IT 14.2

Multiply: $\left(7x-y\right)\left(2x-5\right)$ .

Show answer

$14{x}^{2}-35x-2xy+5y$

Be careful of the exponents in the next example.

EXAMPLE 15

Multiply: $\left({n}^{2}+4\right)\left(n-1\right)$ .

Solution

$\left({n}^{2}+4\right)\left(n-1\right)$
Multiply the First.	$\begin{array}{ccccccc} {\color{red}{n^3}} & + & \rule{0.5cm}{0.4pt} & + & \rule{0.5cm}{0.4pt} &+& \rule{0.5cm}{0.4pt} \\ F&&O&&I&&L \end{array}$
Multiply the Outer.	$\begin{array}{ccccccc} n^3 & {\color{red}{-}} & {\color{red}{n^2}} & + & \rule{0.5cm}{0.4pt} &+& \rule{0.5cm}{0.4pt} \\ F&&O&&I&&L \end{array}$
Multiply the Inner.	$\begin{array}{ccccccc} n^3 & - & n^2 & + & {\color{red}{4n}} &+& \rule{0.5cm}{0.4pt} \\ F&&O&&I&&L \end{array}$
Multiply the Last.	$\begin{array}{ccccccc} n^3 & - & n^2 & + & 4n & {\color{red}{-}}& {\color{red}{4}} \\ F&&O&&I&&L \end{array}$
Combine like terms—there are none.	$n^3-n^2+4 n-4$

TRY IT 15.1

Multiply: $\left({x}^{2}+6\right)\left(x-8\right)$ .

Show answer

${x}^{3}-8{x}^{2}+6x-48$

TRY IT 15.2

Multiply: $\left({y}^{2}+7\right)\left(y-9\right)$ .

Show answer

${y}^{3}-9{y}^{2}+7y-63$

EXAMPLE 16

Multiply: $\left(3pq+5\right)\left(6pq-11\right)$ .

Solution

$(3pq+5)(6pq-11)$
Multiply the First.	$\begin{array}{ccccccc} {\color{red}{18p^2q^2}} & + & \rule{.5cm}{0.4pt} & + & \rule{.5cm}{0.4pt} & + & \rule{.5cm}{0.4pt} \\ F & & O & & I & & L \\ \end{array}$
Multiply the Outer.	$\begin{array}{ccccccc} 18p^2q^2 & {\color{red}{-}} &{\color{red}{33pq}} & + & \rule{.5cm}{0.4pt} & + & \rule{.5cm}{0.4pt} \\ F & & O & & I & & L \\ \end{array}$
Multiply the Inner.	$\begin{array}{ccccccc} 18p^2q^2 & - &33pq & {\color{red}{+}} & {\color{red}{30pq}} & + & \rule{.5cm}{0.4pt} \\ F & & O & & I & & L \\ \end{array}$
Multiply the Last.	$\begin{array}{ccccccc} 18p^2q^2 & - &33pq & + & 30pq & {\color{red}{-}}& {\color{red}{55}} \\ F & & O & & I & & L \\ \end{array}$
Combine like terms.	$18 p^2 q^2-3 p q-55$

TRY IT 16.1

Multiply: $\left(2ab+5\right)\left(4ab-4\right)$ .

Show answer

$8{a}^{2}{b}^{2}+12ab-20$

TRY IT 16.2

Multiply: $\left(2xy+3\right)\left(4xy-5\right)$ .

Show answer

$8{x}^{2}{y}^{2}+2xy-15$

Multiply a Binomial by a Binomial Using the Vertical Method

The FOIL method is usually the quickest method for multiplying two binomials, but it only works for binomials. You can use the Distributive Property to find the product of any two polynomials. Another method that works for all polynomials is the Vertical Method. It is very much like the method you use to multiply whole numbers. Look carefully at this example of multiplying two-digit numbers.

Now we’ll apply this same method to multiply two binomials.

EXAMPLE 17

Multiply using the Vertical Method: $\left(3y-1\right)\left(2y-6\right)$ .

Solution

It does not matter which binomial goes on the top.

Setup for Vertical multiplication	$\begin{array}{r} 3y-1 \\ \times \quad {\color{blue}{ 2y}}- {\color{red}{6}} \\ \hline \end{array}$
Multiply $3y-1$ by $\color{red} -6$ Multiple $3y-1$ by $\color{blue} 2y$	$\begin{array}{r} {\color{red}{-18y+6}} \\ {\color{blue}{6{y}^{2} -2y}} \hspace{0.9cm} \\ \hline \end{array}$	Partial Product $-18y+6$ Partial Product $6{y}^{2} -2y$
Add like terms.	$6y^2 \hspace{0.2cm} -20 y \hspace{0.2cm} +6$	Product $6{y}^{2} - 20y+6$

If you use the FOIL method to multiply these binomials you will notice that the partial products are the same as the terms in the FOIL method.

TRY IT 17.1

Multiply using the Vertical Method: $\left(5m-7\right)\left(3m-6\right)$ .

Show answer

$15{m}^{2}-51m+42$

TRY IT 17.2

Multiply using the Vertical Method: $\left(6b-5\right)\left(7b-3\right)$ .

Show answer

$42{b}^{2}-53b+15$

We have now used three methods for multiplying binomials. Be sure to practice each method, and try to decide which one you prefer. The methods are listed here all together, to help you remember them.

HOW TO: Multiplying Two Binomials

To multiply binomials, use the:

- Distributive Property
- FOIL Method
- Vertical Method

Remember, FOIL only works when multiplying two binomials.

Multiply a Trinomial by a Binomial

We have multiplied monomials by monomials, monomials by polynomials, and binomials by binomials. Now we’re ready to multiply a trinomial by a binomial. Remember, FOIL will not work in this case, but we can use either the Distributive Property or the Vertical Method. We first look at an example using the Distributive Property.

EXAMPLE 18

Multiply using the Distributive Property: $\left(b+3\right)\left(2{b}^{2}-5b+8\right)$ .

Solution


Distribute.
Multiply.	$2 b^3-5b^2+8b+6b^2-15 b+24$
Combine like terms.	$2 b^3+b^2-7 b+24$

TRY IT 18.1

Multiply using the Distributive Property: $\left(y-3\right)\left({y}^{2}-5y+2\right)$ .

Show answer

${y}^{3}-8{y}^{2}+17y-6$

TRY IT 18.2

Multiply using the Distributive Property: $\left(x+4\right)\left(2{x}^{2}-3x+5\right)$ .

Show answer

$2{x}^{3}+5{x}^{2}-7x+20$

Now let’s do this same multiplication using the Vertical Method.

EXAMPLE 19

Multiply using the Vertical Method: $\left(b+3\right)\left(2{b}^{2}-5b+8\right)$ .

Solution

It is easier to put the polynomial with fewer terms on the bottom because we get fewer partial products this way.

Rewrite the question in vertical setup	$\begin{array}{r} 2 b^2-5 b+8 \\ \times \quad {\color{blue}{ b}}+ {\color{red}{3}} \\ \hline \end{array}$
Multiply (2b² − 5b + 8) by $\color{red}3$ . Multiply (2b² − 5b + 8) by $\color{blue}b$ .	$\begin{array}{r} {\color{red}{6 b^2-15 b+24}} \\ {\color{blue}{2b^3-5b^2+8b}} \hspace{0.9cm} \\ \hline \end{array}$
Add like terms.	$2 b^3 +b^2 \hspace{0.2cm} -7 b \hspace{0.2cm} +24$
There are no more like terms that can be collected.	$2 b^3+b^2-7 b+24$

TRY IT 19.1

Multiply using the Vertical Method: $\left(y-3\right)\left({y}^{2}-5y+2\right)$ .

Show answer

${y}^{3}-8{y}^{2}+17y-6$

TRY IT 19.2

Multiply using the Vertical Method: $\left(x+4\right)\left(2{x}^{2}-3x+5\right)$ .

Show answer

$2{x}^{3}+5{x}^{2}-7x+20$

We have now seen two methods you can use to multiply a trinomial by a binomial. After you practice each method, you’ll probably find you prefer one way over the other. We list both methods are listed here, for easy reference.

HOW TO: Multiply a Trinomial by a Binomial

To multiply a trinomial by a binomial, use the:

Distributive Property
Vertical Method

Access these online resources for additional instruction and practice with multiplying polynomials:

Key Concepts

FOIL Method for Multiplying Two Binomials—To multiply two binomials:
1. Multiply the First terms.
2. Multiply the Outer terms.
3. Multiply the Inner terms.
4. Multiply the Last terms.
Multiplying Two Binomials—To multiply binomials, use the:
- Distributive Property ((Figure))
- FOIL Method ((Figure))
Multiplying a Trinomial by a Binomial—To multiply a trinomial by a binomial, use the:
- Distributive Property ((Figure))

Practice Makes Perfect

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

1. $4\left(w+10\right)$	2. $6\left(b+8\right)$
3. $-3\left(a+7\right)$	4. $-5\left(p+9\right)$
5. $2\left(x-7\right)$	6. $7\left(y-4\right)$
7. $-3\left(k-4\right)$	8. $-8\left(j-5\right)$
9. $q\left(q+5\right)$	10. $k\left(k+7\right)$
11. $-b\left(b+9\right)$	12. $-y\left(y+3\right)$
13. $-x\left(x-10\right)$	14. $-p\left(p-15\right)$
15. $6r\left(4r+s\right)$	16. $5c\left(9c+d\right)$
17. $12x\left(x-10\right)$	18. $9m\left(m-11\right)$
19. $-9a\left(3a+5\right)$	20. $-4p\left(2p+7\right)$
21. $3\left({p}^{2}+10p+25\right)$	22. $6\left({y}^{2}+8y+16\right)$
23. $-8x\left({x}^{2}+2x-15\right)$	24. $-5t\left({t}^{2}+3t-18\right)$
25. $5{q}^{3}\left({q}^{3}-2q+6\right)$	26. $4{x}^{3}\left({x}^{4}-3x+7\right)$
27. $-8y\left({y}^{2}+2y-15\right)$	28. $-5m\left({m}^{2}+3m-18\right)$
29. $5{q}^{3}\left({q}^{2}-2q+6\right)$	30. $9{r}^{3}\left({r}^{2}-3r+5\right)$
31. $-4{z}^{2}\left(3{z}^{2}+12z-1\right)$	32. $-3{x}^{2}\left(7{x}^{2}+10x-1\right)$
33. $\left(2m-9\right)m$	34. $\left(8j-1\right)j$
35. $\left(w-6\right)\cdot 8$	36. $\left(k-4\right)\cdot 5$
37. $4\left(x+10\right)$	38. $6\left(y+8\right)$
39. $15\left(r-24\right)$	40. $12\left(v-30\right)$
41. $-3\left(m+11\right)$	42. $-4\left(p+15\right)$
43. $-8\left(z-5\right)$	44. $-3\left(x-9\right)$
45. $u\left(u+5\right)$	46. $q\left(q+7\right)$
47. $n\left({n}^{2}-3n\right)$	48. $s\left({s}^{2}-6s\right)$
49. $6x\left(4x+y\right)$	50. $5a\left(9a+b\right)$
51. $5p\left(11p-5q\right)$	52. $12u\left(3u-4v\right)$
53. $3\left({v}^{2}+10v+25\right)$	54. $6\left({x}^{2}+8x+16\right)$
55. $2n\left(4{n}^{2}-4n+1\right)$	56. $3r\left(2{r}^{2}-6r+2\right)$
57. $-8y\left({y}^{2}+2y-15\right)$	58. $5m\left({m}^{2}+3m+8\right)$
59. $5{q}^{3}\left({q}^{2}-2q+6\right)$	60. $9{r}^{3}\left({r}^{2}-3r+5\right)$
61. $-4{z}^{2}\left(3{z}^{2}+12z-1\right)$	62. $-3{x}^{2}\left(7{x}^{2}+10x-1\right)$
63. $\left(2y-9\right)y$	64. $\left(8b-1\right)b$

Multiply a Binomial by a Binomial

In the following exercises, multiply the following binomials using: a) the Distributive Property b) the FOIL method c) the Vertical Method.

65. $\left(w+5\right)\left(w+7\right)$	66. $\left(y+9\right)\left(y+3\right)$
67. $\left(p+11\right)\left(p-4\right)$	68. $\left(q+4\right)\left(q-8\right)$

In the following exercises, multiply the binomials. Use any method.

69. $\left(x+8\right)\left(x+3\right)$	70. $\left(y+7\right)\left(y+4\right)$
71. $\left(y-6\right)\left(y-2\right)$	72. $\left(x-7\right)\left(x-2\right)$
73. $\left(w-4\right)\left(w+7\right)$	74. $\left(q-5\right)\left(q+8\right)$
75. $\left(p+12\right)\left(p-5\right)$	76. $\left(m+11\right)\left(m-4\right)$
77. $\left(6p+5\right)\left(p+1\right)$	78. $\left(7m+1\right)\left(m+3\right)$
79. $\left(2t-9\right)\left(10t+1\right)$	80. $\left(3r-8\right)\left(11r+1\right)$
81. $\left(5x-y\right)\left(3x-6\right)$	82. $\left(10a-b\right)\left(3a-4\right)$
83. $\left(a+b\right)\left(2a+3b\right)$	84. $\left(r+s\right)\left(3r+2s\right)$
85. $\left(4z-y\right)\left(z-6\right)$	86. $\left(5x-y\right)\left(x-4\right)$
87. $\left({x}^{2}+3\right)\left(x+2\right)$	88. $\left({y}^{2}-4\right)\left(y+3\right)$
89. $\left({x}^{2}+8\right)\left({x}^{2}-5\right)$	90. $\left({y}^{2}-7\right)\left({y}^{2}-4\right)$
91. $\left(5ab-1\right)\left(2ab+3\right)$	92. $\left(2xy+3\right)\left(3xy+2\right)$
93. $\left(6pq-3\right)\left(4pq-5\right)$	94. $\left(3rs-7\right)\left(3rs-4\right)$

Multiply a Trinomial by a Binomial

In the following exercises, multiply using a) the Distributive Property b) the Vertical Method.

95. $\left(x+5\right)\left({x}^{2}+4x+3\right)$	96. $\left(u+4\right)\left({u}^{2}+3u+2\right)$
97. $\left(y+8\right)\left(4{y}^{2}+y-7\right)$	98. $\left(a+10\right)\left(3{a}^{2}+a-5\right)$

In the following exercises, multiply. Use either method.

99. $\left(w-7\right)\left({w}^{2}-9w+10\right)$	100. $\left(p-4\right)\left({p}^{2}-6p+9\right)$
101. $\left(3q+1\right)\left({q}^{2}-4q-5\right)$	102. $\left(6r+1\right)\left({r}^{2}-7r-9\right)$

Mixed Practice

103. $\left(10y-6\right)+\left(4y-7\right)$	104. $\left(15p-4\right)+\left(3p-5\right)$
105. $\left({x}^{2}-4x-34\right)-\left({x}^{2}+7x-6\right)$	106. $\left({j}^{2}-8j-27\right)-\left({j}^{2}+2j-12\right)$
107. $5q\left(3{q}^{2}-6q+11\right)$	108. $8t\left(2{t}^{2}-5t+6\right)$
109. $\left(s-7\right)\left(s+9\right)$	110. $\left(x-5\right)\left(x+13\right)$
111. $\left({y}^{2}-2y\right)\left(y+1\right)$	112. $\left({a}^{2}-3a\right)\left(4a+5\right)$
113. $\left(3n-4\right)\left({n}^{2}+n-7\right)$	114. $\left(6k-1\right)\left({k}^{2}+2k-4\right)$
115. $\left(7p+10\right)\left(7p-10\right)$	116. $\left(3y+8\right)\left(3y-8\right)$
117. $\left(4{m}^{2}-3m-7\right){m}^{2}$	118. $\left(15{c}^{2}-4c+5\right){c}^{4}$
119. $\left(5a+7b\right)\left(5a+7b\right)$	120. $\left(3x-11y\right)\left(3x-11y\right)$
121. $\left(4y+12z\right)\left(4y-12z\right)$

Everyday Math

122. Mental math You can use binomial multiplication to multiply numbers without a calculator. Say you need to multiply 13 times 15. Think of 13 as $10+3$ and 15 as $10+5$ .

Multiply $\left(10+3\right)\left(10+5\right)$ by the FOIL method.
Multiply $13\cdot 15$ without using a calculator.
Which way is easier for you? Why?

123. Mental math You can use binomial multiplication to multiply numbers without a calculator. Say you need to multiply 18 times 17. Think of 18 as $20-2$ and 17 as $20-3$ .

Multiply $\left(20-2\right)\left(20-3\right)$ by the FOIL method.
Multiply $18\cdot 17$ without using a calculator.
Which way is easier for you? Why?

Writing Exercises

124. Which method do you prefer to use when multiplying two binomials: the Distributive Property, the FOIL method, or the Vertical Method? Why?

125. Which method do you prefer to use when multiplying a trinomial by a binomial: the Distributive Property or the Vertical Method? Why?

126. Multiply the following:

$\begin{array}{c}\left(x+2\right)\left(x-2\right)\hfill \\ \left(y+7\right)\left(y-7\right)\hfill \\ \left(w+5\right)\left(w-5\right)\hfill \end{array}$

Explain the pattern that you see in your answers.

127. Multiply the following:

$\begin{array}{c}\left(m-3\right)\left(m+3\right)\hfill \\ \left(n-10\right)\left(n+10\right)\hfill \\ \left(p-8\right)\left(p+8\right)\hfill \end{array}$

Explain the pattern that you see in your answers.

128. Multiply the following:

$\begin{array}{c}\left(p+3\right)\left(p+3\right)\hfill \\ \left(q+6\right)\left(q+6\right)\hfill \\ \left(r+1\right)\left(r+1\right)\hfill \end{array}$

Explain the pattern that you see in your answers.

129. Multiply the following:

$\begin{array}{c}\left(x-4\right)\left(x-4\right)\hfill \\ \left(y-1\right)\left(y-1\right)\hfill \\ \left(z-7\right)\left(z-7\right)\hfill \end{array}$

Explain the pattern that you see in your answers.

Answers

1. $4w+40$	3. $-3a-21$
5. $2x-14$	7. $-3k+12$
9. ${q}^{2}+5q$	11. $\text{-}{b}^{2}-9b$
13. $\text{-}{x}^{2}+10x$	15. $24{r}^{2}+6rs$
17. $12{x}^{2}-120x$	19. $-27{a}^{2}-45a$
21. $3{p}^{2}+30p+75$	23. $-8{x}^{3}-16{x}^{2}+120x$
25. $5{q}^{6}-10{q}^{4}+30{q}^{3}$	27. $-8{y}^{3}-16{y}^{2}+120y$
29. $5{q}^{5}-10{q}^{4}+30{q}^{3}$	31. $-12{z}^{4}-48{z}^{3}+4{z}^{2}$
33. $2{m}^{2}-9m$	35. $8w-48$
37. $4x+40$	39. $15r-360$
41. $-3m-33$	43. $-8z+40$
45. ${u}^{2}+5u$	47. ${n}^{3}-3{n}^{2}$
49. $24{x}^{2}+6xy$	51. $55{p}^{2}-25pq$
53. $3{v}^{2}+30v+75$	55. $8{n}^{3}-8{n}^{2}+2n$
57. $-8{y}^{3}-16{y}^{2}+120y$	59. $5{q}^{5}-10{q}^{4}+30{q}^{3}$
61. $-12{z}^{4}-48{z}^{3}+4{z}^{2}$	63. $2{y}^{2}-9y$
65. ${w}^{2}+12w+35$	67. ${p}^{2}+7p-44$
69. ${x}^{2}+11x+24$	71. ${y}^{2}-8y+12$
73. ${w}^{2}+3w-28$	75. ${p}^{2}+7p-60$
77. $6{p}^{2}+11p+5$	79. $20{t}^{2}-88t-9$
81. $15{x}^{2}-3xy-30x+6y$	83. $2{a}^{2}+5ab+3{b}^{2}$
85. $4{z}^{2}-24z-zy+6y$	87. ${x}^{3}+2{x}^{2}+3x+6$
89. ${x}^{4}+3{x}^{2}-40$	91. $10{a}^{2}{b}^{2}+13ab-3$
93. $24{p}^{2}{q}^{2}-42pq+15$	95. ${x}^{3}+9{x}^{2}+23x+15$
97. $4{y}^{3}+33{y}^{2}+y-56$	99. ${w}^{3}-16{w}^{2}+73w-70$
101. $3{q}^{3}-11{q}^{2}-19q-5$	103. $14y-13$
105. $-11x-28$	107. $15{q}^{3}-30{q}^{2}+55q$
109. ${s}^{2}+2s-63$	111. ${y}^{3}-{y}^{2}-2y$
113. $3{n}^{3}-{n}^{2}-25n+28$	115. $49{p}^{2}-100$
117. $4{m}^{4}-3{m}^{3}-7{m}^{2}$	119. $25{a}^{2}+70ab+49{b}^{2}$
121. $16{y}^{2}-144{z}^{2}$	123. a) 306 b) 306 c) Answers will vary.
125. Answers will vary.	127. ${m}^2-9$ ${n}^2-100$ ${p}^2-64$ Answers will vary.
129. ${x}^2-8x+16$ ${y}^2-2y+1$ ${z}^2-14z+49$ Answers will vary.

Attributions

This chapter has been adapted from “Multiply Polynomials” in Prealgebra (OpenStax) by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

License

Icon for the Creative Commons Attribution 4.0 International License

Multiply a Polynomial by a Monomial

Multiply a Binomial by a Binomial

Multiply a Binomial by a Binomial Using the Distributive Property

Multiply a Binomial by a Binomial Using the FOIL Method

Multiply a Binomial by a Binomial Using the Vertical Method

Multiply a Trinomial by a Binomial

Key Concepts

Practice Makes Perfect

Multiply a Polynomial by a Monomial

Multiply a Binomial by a Binomial

Multiply a Trinomial by a Binomial

Mixed Practice

Everyday Math

Writing Exercises

Answers

Attributions

License

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