Multiply Polynomials

Multiply a Polynomial by a Monomial

We have used the Distributive Property to simplify expressions like $2 (x - 3)$ . You multiplied both terms in the parentheses, $x and 3$ , by 2, to get $2 x - 6$ . 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:

Multiply: $4 (x + 3) .$

Solution


Distribute.
Simplify.

Multiply: $y (y - 2) .$

Solution


Distribute.
Simplify.

Multiply: $7 x (2 x + y) .$

Solution


Distribute.
Simplify.

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

Solution


Distribute.
Simplify.

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

Solution


Distribute.
Simplify.

Multiply: $(x + 3) p .$

Solution

The monomial is the second factor.
Distribute.
Simplify.

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 Example 6, where we multiplied a binomial by a monomial.


We distributed the p to get:
What if we have (x + 7) instead of p?
Distribute (x + 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.

Multiply: $(y + 5) (y + 8) .$

Solution


Distribute (y + 8).
Distribute again
Combine like terms.

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

Solution


Distribute (3y + 4).
Distribute again
Combine like terms.

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

Solution


Distribute.
Distribute again.
Combine like terms.

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

Solution


Distribute.
Distribute again.
There are no like terms to combine.

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 Example 10, 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{matrix} (x - 2) (x - y) \\ x^{2} - x y - 2 x + 2 y \end{matrix}

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

This figure explains how to multiply a binomial using the FOIL method. It has two columns, with written instructions on the left and math on the right. At the top of the figure, the text in the left column says “It is the product of x and x, the first terms in x minus 2 and x minus y.” In the right column is the product of x minus 2 and x minus y. An arrow extends from the x in x minus 2, and terminates at the x in x minus y. Below this is the word “First.” One row down, the text in the left column says “The next terms, negative xy, is the product of x and negative y, the two outer terms.” In the right column is the product of x minus 2 and x minus y, with another arrow extending from the x in x minus 2 to the y in x minus y. Below this is the word “Outer.” One row down, the text in the left column says “The third term, negative 2 x, is the product of negative 2 and x, the two inner terms.” In the right column is the product of x minus 2 and x minus y with a third arrow extending from minus 2 in x minus 2 and terminating at the x in x minus y. Below this is the word “Inner.” In the last row, the text in the left column says “And the last term, plus 2y, came from multiplying the two last terms, negative 2 and negative y.” In the right column is the product of x minus 2 and x minus y, with a fourth arrow extending from the minus 2 in x minus 2 to the minus y in x minus y. Below this is the word “Last.”

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.

\begin{matrix} (x - 2) (x - y) \\ \underset{F O I L}{x^{2} - x y - 2 x + 2 y} \end{matrix}

Let’s look at $(x + 3) (x + 7)$ .

Distibutive Property	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.

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

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

Solution

This figure is a table that has three columns and five rows. The first column is a header column, and it contains the names and numbers of each step. The second and third columns contain math. On the top row of the table, the first cell on the left reads “Step 1. Multiply the first terms.” The second column contains the product of binomials x plus 5 and x plus 9. Below this is the product of x plus 5 and x plus 9 again, with an arrow extending from the x in the first binomial to the x in the second binomial. The third column contains x squared plus blank plus blank plus blank. Below the x squared is the letter F, and below each of the three blanks are the letters O, I, and L, respectively.

In the second row, the first cell reads “Step 2. Multiply the outer terms.” In the second cell is the product of x plus 5 and x plus 9 again, with an arrow extending from x in the first binomial to the 9 in the second binomial. The third cell contains x squared plus 9x plus blank plus blank, with the letter F under the x squared, O under the 9x, and I and L beneath the two blanks.

In the third row, the first cell reads “Step 3. Multiply the inner terms.” The second cell contains the product of x plus 5 and x plus 9 again, with an arrow extending from 5 in the first binomial to the x in the second binomial. The third cell contains x squared plus 9x plus 5x plus blank, with F beneath x squared, O beneath 9x, I beneath 5x, and L beneath the blank.

In the fourth row, the first cell reads “Step 4. Multiply the last terms.” In the second cell is the product of x plus 5 and x plus 9 again, with an arrow extending from 5 in the first binomial to 9 in the second binomial. The third cell contains x squared plus 9x plus 6x plus 45, with F beneath x squared, O beneath 9x, I beneath 6x, and L beneath 45.

In the final row, the first cell reads “Step 5. Combine like terms, when possible.” The second cell is blank. The third cell contains the final expression: x squared plus 15x plus 45.

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

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

Multiply: $(y - 7) (y + 4) .$

Solution

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

Solution

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

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

Solution



Multiply the First.
Multiply the Outer.
Multiply the Inner.
Multiply the Last.
Combine like terms—there are none.

Be careful of the exponents in the next example.

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

Solution



Multiply the First.
Multiply the Outer.
Multiply the Inner.
Multiply the Last.
Combine like terms—there are none.

Multiply: $(3 p q + 5) (6 p q - 11) .$

Solution


Multiply the First.
Multiply the Outer.
Multiply the Inner.
Multiply the Last.
Combine like terms—there are none.

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.

This figure shows the vertical multiplication of 23 and 46. The number 23 is above the number 46. Below this, there is the partial product 138 over the partial product 92. The final product is at the bottom and is 1058. Text on the right side of the image says “Start by multiplying 23 by 6 to get 138. Next, multiply 23 by 4, lining up the partial product in the correct columns. Last you add the partial products.”

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

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

Solution

It does not matter which binomial goes on the top.

$\begin{matrix} \begin{matrix} Multiply 3 y - 1 by −6 . \\ Multiply 3 y - 1 by 2y. \\ Add like terms. \end{matrix} & \begin{matrix} 3 y - 1 \\ \underset{________}{\times 2 y - 6} \\ −18 y + 6 \\ \underset{_____________}{6 y^{2} - 2 y} \\ 6 y^{2} - 20 y + 6 \end{matrix} & \begin{matrix} partial product \\ partial product \\ product \end{matrix} \end{matrix}$

Notice the partial products are the same as the terms in the FOIL method.

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.

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.

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

Solution


Distribute.
Multiply.
Combine like terms.

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

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

Solution

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

Multiply (2b² − 5b + 8) by 3.
Multiply (2b² − 5b + 8) by b.
Add like terms.

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.

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 (Example 7)
- FOIL Method (Example 12)
- Vertical Method (Example 17)
Multiplying a Trinomial by a Binomial—To multiply a trinomial by a binomial, use the:
- Distributive Property (Example 18)
- Vertical Method (Example 19)

Practice Makes Perfect

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

$4 (w + 10)$

Solution

$4 w + 40$

$6 (b + 8)$

$−3 (a + 7)$

Solution

$−3 a - 21$

$−5 (p + 9)$

$2 (x - 7)$

Solution

$2 x - 14$

$7 (y - 4)$

$−3 (k - 4)$

Solution

$−3 k + 12$

$−8 (j - 5)$

$q (q + 5)$

Solution

$q^{2} + 5 q$

$k (k + 7)$

$− b (b + 9)$

Solution

$− b^{2} - 9 b$

$− y (y + 3)$

$− x (x - 10)$

Solution

$− x^{2} + 10 x$

$− p (p - 15)$

$6 r (4 r + s)$

Solution

$24 r^{2} + 6 r s$

$5 c (9 c + d)$

$12 x (x - 10)$

Solution

$12 x^{2} - 120 x$

$9 m (m - 11)$

$−9 a (3 a + 5)$

Solution

$−27 a^{2} - 45 a$

$−4 p (2 p + 7)$

$3 (p^{2} + 10 p + 25)$

Solution

$3 p^{2} + 30 p + 75$

$6 (y^{2} + 8 y + 16)$

$−8 x (x^{2} + 2 x - 15)$

Solution

$−8 x^{3} - 16 x^{2} + 120 x$

$−5 t (t^{2} + 3 t - 18)$

$5 q^{3} (q^{3} - 2 q + 6)$

Solution

$5 q^{6} - 10 q^{4} + 30 q^{3}$

$4 x^{3} (x^{4} - 3 x + 7)$

$−8 y (y^{2} + 2 y - 15)$

Solution

$−8 y^{3} - 16 y^{2} + 120 y$

$−5 m (m^{2} + 3 m - 18)$

$5 q^{3} (q^{2} - 2 q + 6)$

Solution

$5 q^{5} - 10 q^{4} + 30 q^{3}$

$9 r^{3} (r^{2} - 3 r + 5)$

$−4 z^{2} (3 z^{2} + 12 z - 1)$

Solution

$−12 z^{4} - 48 z^{3} + 4 z^{2}$

$−3 x^{2} (7 x^{2} + 10 x - 1)$

$(2 m - 9) m$

Solution

$2 m^{2} - 9 m$

$(8 j - 1) j$

$(w - 6) \cdot 8$

Solution

$8 w - 48$

$(k - 4) \cdot 5$

$4 (x + 10)$

Solution

$4 x + 40$

$6 (y + 8)$

$15 (r - 24)$

Solution

$15 r - 360$

$12 (v - 30)$

$−3 (m + 11)$

Solution

$−3 m - 33$

$−4 (p + 15)$

$−8 (z - 5)$

Solution

$−8 z + 40$

$−3 (x - 9)$

$u (u + 5)$

Solution

$u^{2} + 5 u$

$q (q + 7)$

$n (n^{2} - 3 n)$

Solution

$n^{3} - 3 n^{2}$

$s (s^{2} - 6 s)$

$6 x (4 x + y)$

Solution

$24 x^{2} + 6 x y$

$5 a (9 a + b)$

$5 p (11 p - 5 q)$

Solution

$55 p^{2} - 25 p q$

$12 u (3 u - 4 v)$

$3 (v^{2} + 10 v + 25)$

Solution

$3 v^{2} + 30 v + 75$

$6 (x^{2} + 8 x + 16)$

$2 n (4 n^{2} - 4 n + 1)$

Solution

$8 n^{3} - 8 n^{2} + 2 n$

$3 r (2 r^{2} - 6 r + 2)$

$−8 y (y^{2} + 2 y - 15)$

Solution

$−8 y^{3} - 16 y^{2} + 120 y$

$−5 m (m^{2} + 3 m - 18)$

$5 q^{3} (q^{2} - 2 q + 6)$

Solution

$5 q^{5} - 10 q^{4} + 30 q^{3}$

$9 r^{3} (r^{2} - 3 r + 5)$

$−4 z^{2} (3 z^{2} + 12 z - 1)$

Solution

$−12 z^{4} - 48 z^{3} + 4 z^{2}$

$−3 x^{2} (7 x^{2} + 10 x - 1)$

$(2 y - 9) y$

Solution

$2 y^{2} - 9 y$

$(8 b - 1) b$

Multiply a Binomial by a Binomial

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

$(w + 5) (w + 7)$

Solution

$w^{2} + 12 w + 35$

$(y + 9) (y + 3)$

$(p + 11) (p - 4)$

Solution

$p^{2} + 7 p - 44$

$(q + 4) (q - 8)$

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

$(x + 8) (x + 3)$

Solution

$x^{2} + 11 x + 24$

$(y + 7) (y + 4)$

$(y - 6) (y - 2)$

Solution

$y^{2} - 8 y + 12$

$(x - 7) (x - 2)$

$(w - 4) (w + 7)$

Solution

$w^{2} + 3 w - 28$

$(q - 5) (q + 8)$

$(p + 12) (p - 5)$

Solution

$p^{2} + 7 p - 60$

$(m + 11) (m - 4)$

$(6 p + 5) (p + 1)$

Solution

$6 p^{2} + 11 p + 5$

$(7 m + 1) (m + 3)$

$(2 t - 9) (10 t + 1)$

Solution

$20 t^{2} - 88 t - 9$

$(3 r - 8) (11 r + 1)$

$(5 x - y) (3 x - 6)$

Solution

$15 x^{2} - 3 x y - 30 x + 6 y$

$(10 a - b) (3 a - 4)$

$(a + b) (2 a + 3 b)$

Solution

$2 a^{2} + 5 a b + 3 b^{2}$

$(r + s) (3 r + 2 s)$

$(4 z - y) (z - 6)$

Solution

$4 z^{2} - 24 z - z y + 6 y$

$(5 x - y) (x - 4)$

$(x^{2} + 3) (x + 2)$

Solution

$x^{3} + 2 x^{2} + 3 x + 6$

$(y^{2} - 4) (y + 3)$

$(x^{2} + 8) (x^{2} - 5)$

Solution

$x^{4} + 3 x^{2} - 40$

$(y^{2} - 7) (y^{2} - 4)$

$(5 a b - 1) (2 a b + 3)$

Solution

$10 a^{2} b^{2} + 13 a b - 3$

$(2 x y + 3) (3 x y + 2)$

$(6 p q - 3) (4 p q - 5)$

Solution

$24 p^{2} q^{2} - 42 p q + 15$

$(3 r s - 7) (3 r s - 4)$

Multiply a Trinomial by a Binomial

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

$(x + 5) (x^{2} + 4 x + 3)$

Solution

$x^{3} + 9 x^{2} + 23 x + 15$

$(u + 4) (u^{2} + 3 u + 2)$

$(y + 8) (4 y^{2} + y - 7)$

Solution

$4 y^{3} + 33 y^{2} + y - 56$

$(a + 10) (3 a^{2} + a - 5)$

In the following exercises, multiply. Use either method.

$(w - 7) (w^{2} - 9 w + 10)$

Solution

$w^{3} - 16 w^{2} + 73 w - 70$

$(p - 4) (p^{2} - 6 p + 9)$

$(3 q + 1) (q^{2} - 4 q - 5)$

Solution

$3 q^{3} - 11 q^{2} - 19 q - 5$

$(6 r + 1) (r^{2} - 7 r - 9)$

Mixed Practice

$(10 y - 6) + (4 y - 7)$

Solution

$14 y - 13$

$(15 p - 4) + (3 p - 5)$

$(x^{2} - 4 x - 34) - (x^{2} + 7 x - 6)$

Solution

$−11 x - 28$

$(j^{2} - 8 j - 27) - (j^{2} + 2 j - 12)$

$5 q (3 q^{2} - 6 q + 11)$

Solution

$15 q^{3} - 30 q^{2} + 55 q$

$8 t (2 t^{2} - 5 t + 6)$

$(s - 7) (s + 9)$

Solution

$s^{2} + 2 s - 63$

$(x - 5) (x + 13)$

$(y^{2} - 2 y) (y + 1)$

Solution

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

$(a^{2} - 3 a) (4 a + 5)$

$(3 n - 4) (n^{2} + n - 7)$

Solution

$3 n^{3} - n^{2} - 25 n + 28$

$(6 k - 1) (k^{2} + 2 k - 4)$

$(7 p + 10) (7 p - 10)$

Solution

$49 p^{2} - 100$

$(3 y + 8) (3 y - 8)$

$(4 m^{2} - 3 m - 7) m^{2}$

Solution

$4 m^{4} - 3 m^{3} - 7 m^{2}$

$(15 c^{2} - 4 c + 5) c^{4}$

$(5 a + 7 b) (5 a + 7 b)$

Solution

$25 a^{2} + 70 a b + 49 b^{2}$

$(3 x - 11 y) (3 x - 11 y)$

$(4 y + 12 z) (4 y - 12 z)$

Solution

$16 y^{2} - 144 z^{2}$

Everyday Math

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 $(10 + 3) (10 + 5)$ by the FOIL method.
ⓑ Multiply $13 \cdot 15$ without using a calculator.

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 $(20 - 2) (20 - 3)$ by the FOIL method.
ⓑ Multiply $18 \cdot 17$ without using a calculator.

Solution

Writing Exercises

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

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

Solution

Answers will vary.

Multiply the following:

$\begin{matrix} (x + 2) (x - 2) \\ (y + 7) (y - 7) \\ (w + 5) (w - 5) \end{matrix}$

Explain the pattern that you see in your answers.

Multiply the following:

$\begin{matrix} (m - 3) (m + 3) \\ (n - 10) (n + 10) \\ (p - 8) (p + 8) \end{matrix}$

Explain the pattern that you see in your answers.

Solution

Answers may vary.

Multiply the following:

$\begin{matrix} (p + 3) (p + 3) \\ (q + 6) (q + 6) \\ (r + 1) (r + 1) \end{matrix}$

Explain the pattern that you see in your answers.

Multiply the following:

$\begin{matrix} (x - 4) (x - 4) \\ (y - 1) (y - 1) \\ (z - 7) (z - 7) \end{matrix}$

Explain the pattern that you see in your answers.

Solution

Answers may vary.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This is a table that has four rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “multiply a polynomial by a monomial,” “multiply a binomial by a binomial,” and “multiply a trinomial by a binomial.” The rest of the cells are blank.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?