Multiply Polynomials

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

You already know how to use the Distributive Property to simplify expressions like $2(x - 3)$ . 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!

Example. Multiply: $3(x + 7)$ .

Distribute the $3$ , multiplying it by each term inside the parentheses:

3(x + 7) = 3 \cdot x + 3 \cdot 7 = 3x + 21

Multiply: $6(x + 8)$ .

Multiply: $2(y + 12)$ .

Example. Multiply: $x(x - 8)$ .

x(x - 8) = x \cdot x - x \cdot 8 = x^2 - 8x

Multiply: $y(y - 9)$ .

Example. Multiply: $10x(4x + y)$ .

10x(4x + y) = 10x \cdot 4x + 10x \cdot y = 40x^2 + 10xy

Multiply: $8x(x + 3y)$ .

Multiplying a monomial by a trinomial works in much the same way.

Example. Multiply: $-2x(5x^2 + 7x - 3)$ .

-2x(5x^2 + 7x - 3) = (-2x) \cdot 5x^2 + (-2x) \cdot 7x - (-2x) \cdot 3 = -10x^3 - 14x^2 + 6x

Multiply: $-4y(8y^2 + 5y - 9)$ .

Multiply: $3x^2(4x^2 - 3x + 9)$ .

Now consider a monomial as the second factor.

Example. Multiply: $(x + 3)p$ .

(x + 3)p = x \cdot p + 3 \cdot p = xp + 3p

Multiply: $(x + 8)p$ .

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.

Using the Distributive Property

We start by using the Distributive Property. Look again at $(x + 3)p$ from the last example: we distributed the $p$ to get $xp + 3p$ . What if we have $(x + 7)$ instead of $p$ ? Think of the $(x + 7)$ as the $p$ above:

(x + 3)(x + 7)

Distribute $(x + 7)$ :

x(x + 7) + 3(x + 7)

Distribute again:

x^2 + 7x + 3x + 21

Combine like terms:

x^2 + 10x + 21

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

Be careful to distinguish between a sum and a product: $x + x$ is a sum that simplifies to $2x$ by combining like terms, while $x \cdot x$ is a product that simplifies to $x^2$ by adding the exponents of like bases.

Example. Multiply: $(x + 6)(x + 8)$ .

Distribute $(x + 8)$ :

x(x + 8) + 6(x + 8)

Distribute again:

x^2 + 8x + 6x + 48

Simplify:

x^2 + 14x + 48

Multiply using the Distributive Property: $(x + 8)(x + 9)$ .

Now consider binomials where the variable has a coefficient.

Example. Multiply: $(2x + 9)(3x + 4)$ .

Distribute $(3x + 4)$ :

2x(3x + 4) + 9(3x + 4)

Distribute again and simplify:

6x^2 + 8x + 27x + 36 = 6x^2 + 35x + 36

Multiply using the Distributive Property: $(5x + 9)(4x + 3)$ .

In the previous examples, the binomials were sums. When there are differences, pay special attention to make sure the signs of the product are correct.

Example. Multiply: $(4y + 3)(6y - 5)$ .

Distribute $(6y - 5)$ :

4y(6y - 5) + 3(6y - 5)

Distribute again and simplify:

24y^2 - 20y + 18y - 15 = 24y^2 - 2y - 15

Multiply using the Distributive Property: $(7y + 1)(8y - 3)$ .

Up to this point, the product of two binomials has been a trinomial. This is not always the case.

Example. Multiply: $(x + 2)(x - y)$ .

Distribute $(x - y)$ :

x(x - y) + 2(x - y)

Distribute again:

x^2 - xy + 2x - 2y

There are no like terms to combine, so this is the simplified product.

Multiply using the Distributive Property: $(x + 5)(x - y)$ .

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 there are no like terms to combine. Consider $(x + 2)(x - y)$ again, and pay particular attention to how we got the four terms:

(x + 2)(x - y) = x^2 - xy + 2x - 2y

Where did the first term, $x^2$ , come from? It is the product of $x$ and $x$ , the first terms in $(x + 2)$ and $(x - y)$ .

The next term, $-xy$ , is the product of $x$ and $-y$ , the two outer terms.

The third term, $+2x$ , is the product of $2$ and $x$ , the two inner terms.

And the last term, $-2y$ , came from multiplying the two last terms.

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

\underbrace{(a + b)}_{\text{first}}\ \underbrace{(c + d)}_{\text{first}}

The letters remind us of the pattern: first terms, outer terms, inner terms, last terms.

Example. Multiply using the FOIL method: $(x + 6)(x + 9)$ .

Step	Product	Running total
Multiply the First terms	$x \cdot x$	$x^2$
Multiply the Outer terms	$x \cdot 9$	$x^2 + 9x$
Multiply the Inner terms	$6 \cdot x$	$x^2 + 9x + 6x$
Multiply the Last terms	$6 \cdot 9$	$x^2 + 9x + 6x + 54$
Combine like terms		$x^2 + 15x + 54$

Multiply using the FOIL method: $(x + 7)(x + 8)$ .

Multiply using the FOIL method: $(y + 14)(y + 2)$ .

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

Use the FOIL method for multiplying two binomials.

Multiply the First terms.
Multiply the Outer terms.
Multiply the Inner terms.
Multiply the Last terms.
Combine like terms, when possible.

Example. Multiply: $(y - 8)(y + 6)$ .

Step	Product	Running total
First	$y \cdot y$	$y^2$
Outer	$y \cdot 6$	$y^2 + 6y$
Inner	$-8 \cdot y$	$y^2 + 6y - 8y$
Last	$-8 \cdot 6$	$y^2 + 6y - 8y - 48$
Combine like terms		$y^2 - 2y - 48$

Multiply: $(y - 3)(y + 8)$ .

Multiply: $(q - 4)(q + 5)$ .

Example. Multiply: $(2a + 3)(3a - 1)$ .

Step	Product	Running total
First: $2a \cdot 3a$	$6a^2$	$6a^2$
Outer: $2a \cdot (-1)$	$-2a$	$6a^2 - 2a$
Inner: $3 \cdot 3a$	$9a$	$6a^2 - 2a + 9a$
Last: $3 \cdot (-1)$	$-3$	$6a^2 - 2a + 9a - 3$
Combine like terms		$6a^2 + 7a - 3$

Multiply: $(4a + 9)(5a - 2)$ .

Multiply: $(7x + 4)(7x - 8)$ .

Example. Multiply: $(5x - y)(2x - 7)$ .

Step	Product	Running total
First	$10x^2$	$10x^2$
Outer	$-35x$	$10x^2 - 35x$
Inner	$-2xy$	$10x^2 - 35x - 2xy$
Last	$+7y$	$10x^2 - 35x - 2xy + 7y$
Combine like terms — there are none		$10x^2 - 35x - 2xy + 7y$

Multiply: $(12x - y)(x - 5)$ .

Multiply: $(6a - b)(2a - 9)$ .

Using the Vertical Method

The FOIL method is usually the quickest method for multiplying two binomials, but it works only 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:

\begin{array}{r} 23 \\ \times 46 \\ \hline 138 \\ 92\phantom{0} \\ \hline 1058 \end{array}

You start by multiplying $23$ by $6$ to get $138$ , a partial product. Then you multiply $23$ by $4$ , lining up the second partial product, $92$ , in the correct (tens) columns. Last, you add the partial products.

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

Example. Multiply using the vertical method: $(5x - 1)(2x - 7)$ .

It does not matter which binomial goes on top. Line up the columns when you multiply, just as when multiplying $23(46)$ :

\begin{array}{r} 2x - 7 \\ \times\ \ 5x - 1 \\ \hline -2x + 7 \\ 10x^2 - 35x\phantom{00} \\ \hline 10x^2 - 37x + 7 \end{array}

First, multiply $2x - 7$ by $-1$ to get the partial product $-2x + 7$ . Then multiply $2x - 7$ by $5x$ to get the partial product $10x^2 - 35x$ , lining it up one column to the left. Add like terms to get the final product, $10x^2 - 37x + 7$ .

Notice the partial products are the same as the four terms you would get from FOIL.

Multiply using the Vertical Method: $(4m - 9)(3m - 7)$ .

Multiply using the Vertical Method: $(6n - 5)(7n - 2)$ .

We have now used three methods for multiplying binomials. Be sure to practice each method, and try to decide which one you prefer.

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 now multiplied monomials by monomials, monomials by polynomials, and binomials by binomials. Now we’re ready to multiply a trinomial by a binomial. Remember, the FOIL method 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. Multiply using the Distributive Property: $(x + 3)(2x^2 - 5x + 8)$ .

Distribute $(2x^2 - 5x + 8)$ :

x(2x^2 - 5x + 8) + 3(2x^2 - 5x + 8)

Multiply:

2x^3 - 5x^2 + 8x + 6x^2 - 15x + 24

Combine like terms:

2x^3 + x^2 - 7x + 24

Multiply using the Distributive Property: $(y - 1)(y^2 - 7y + 2)$ .

Multiply using the Distributive Property: $(x + 2)(3x^2 - 4x + 5)$ .

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

Example. Multiply using the Vertical Method: $(x + 3)(2x^2 - 5x + 8)$ .

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

\begin{array}{r} 2x^2 - 5x + 8 \\ \times\phantom{00} x + 3 \\ \hline 6x^2 - 15x + 24 \\ 2x^3 - 5x^2 + 8x\phantom{00000} \\ \hline 2x^3 + x^2 - 7x + 24 \end{array}

First multiply $2x^2 - 5x + 8$ by $3$ to get $6x^2 - 15x + 24$ . Then multiply $2x^2 - 5x + 8$ by $x$ to get $2x^3 - 5x^2 + 8x$ , lining it up one column to the left. Add like terms to get the final product.

Multiply using the Vertical Method: $(y - 1)(y^2 - 7y + 2)$ .

Multiply using the Vertical Method: $(x + 2)(3x^2 - 4x + 5)$ .

Key terms

FOIL method — a shortcut for multiplying two binomials: multiply the First terms, the Outer terms, the Inner terms, and the Last terms, then combine any like terms. Vertical Method — a way to multiply polynomials that lines up partial products in columns, the same way you multiply multi-digit whole numbers; it works for any two polynomials, not just binomials.

Practice

Multiply a polynomial by a monomial

Multiply: $-8(z - 5)$ .

Multiply: $-9a(3a + 5)$ .

Multiply: $5q^3(q^2 - 2q + 6)$ .

Multiply: $(w - 6) \cdot 8$ .

Multiply a binomial by a binomial

Multiply: $(z - 10)(z - 22)$ .

Multiply: $(2m - 9)(10m + 1)$ .

Multiply: $(4c - 1)(4c + 1)$ .

Multiply: $(5x - y)(x - 4)$ .

Multiply a trinomial by a binomial

Multiply using the Distributive Property: $(a + 10)(3a^2 + a - 5)$ .

Multiply using the Vertical Method: $(a + 10)(3a^2 + a - 5)$ .

Multiply: $(y - 6)(y^2 - 10y + 9)$ .

Multiply: $(2x + 1)(x^2 - 5x - 6)$ .

This section is adapted from Prealgebra 2e, Section 10.3: Multiply Polynomials by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recreated the FOIL step-by-step worked examples and the vertical (columnar) multiplication layouts as tables and typeset math instead of colored annotated equations; omitted the Be Prepared quiz, Media links, and Self Check checklist; converted the practice problems (“Try Its”) into interactive exercises with instant feedback; and adapted selected end-of-section exercises into the interactive Practice block.

Use Multiplication Properties of Exponents Divide Monomials