Contents

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

Be Prepared 6.3

Before you get started, take this readiness quiz.

Distribute: $2 (x + 3) . 2 (x + 3) .$
If you missed this problem, review Example 1.132.
Combine like terms: $x^{2} + 9 x + 7 x + 63 . x^{2} + 9 x + 7 x + 63 .$
If you missed this problem, review Example 1.24.

Multiply a Polynomial by a Monomial

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

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

Example 6.28

Multiply: $4 (x + 3) . 4 (x + 3) .$

Show solution

Solution


Distribute.
Simplify.

Try It 6.55

Multiply: $5 (x + 7) . 5 (x + 7) .$

Try It 6.56

Multiply: $3 (y + 13) . 3 (y + 13) .$

Example 6.29

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

Show solution

Solution


Distribute.
Simplify.

Try It 6.57

Multiply: $x (x - 7) . x (x - 7) .$

Try It 6.58

Multiply: $d (d - 11) . d (d - 11) .$

Example 6.30

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

Show solution

Solution


Distribute.
Simplify.

Try It 6.59

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

Try It 6.60

Multiply: $2 p (6 p + r) . 2 p (6 p + r) .$

Example 6.31

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

Show solution

Solution


Distribute.
Simplify.

Try It 6.61

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

Try It 6.62

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

Example 6.32

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

Show solution

Solution


Distribute.
Simplify.

Try It 6.63

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

Try It 6.64

Multiply: $−6 a^{3} (3 a^{2} - 2 a + 6) . −6 a^{3} (3 a^{2} - 2 a + 6) .$

Example 6.33

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

Show solution

Solution

The monomial is the second factor.
Distribute.
Simplify.

Try It 6.65

Multiply: $(x + 8) p . (x + 8) p .$

Try It 6.66

Multiply: $(a + 4) p . (a + 4) p .$

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. We will start by using the Distributive Property.

Multiply a Binomial by a Binomial Using the Distributive Property

Look at Example 6.33, 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.

Example 6.34

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

Show solution

Solution


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

Try It 6.67

Multiply: $(x + 8) (x + 9) . (x + 8) (x + 9) .$

Try It 6.68

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

Example 6.35

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

Show solution

Solution


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

Try It 6.69

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

Try It 6.70

Multiply: $(a + 10) (a + 7) . (a + 10) (a + 7) .$

Example 6.36

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

Show solution

Solution


Distribute.
Distribute again.
Combine like terms.

Try It 6.71

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

Try It 6.72

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

Example 6.37

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

Show solution

Solution


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

Try It 6.73

Multiply: $(a + 7) (a - b) . (a + 7) (a - b) .$

Try It 6.74

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

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 6.37, 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} \begin{matrix} (x - 2) (x - y) \\ x^{2} - x y - 2 x + 2 y \end{matrix}

Where did the first term, $x^{2} x^{2}$ , 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} \begin{matrix} (x - 2) (x - y) \\ \underset{F}{x^{2} - x y - 2 x + 2 y} \end{matrix}

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

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.

Example 6.38

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

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

Show solution

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.

Try It 6.75

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

Try It 6.76

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

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 6.39

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

Show solution

Solution

This figure has three columns, with written instructions in the first column and math in the second and third columns. At the top of the figure, the text in the first column says “Multiply the first terms.” The second column contains the product of two binomials, y minus 7 and y plus 4, with an arrow extending from the y in the first binomial to the y in the second binomial. The third column contains y squared plus blank plus blank plus blank. Beneath y squared is the letter F and beneath each blank are the letters O, I, and L, respectively. One row down, the text in the first column says “Multiply the outer terms.” The second column contains the product of y minus 7 and y plus 4 again, with a second arrow extending from y in the first binomial to 4 in the second binomial. The third column contains y squared plus 4y plus blank plus blank. Below y squared is F, below 4y is O, and below the blanks are I and L. One row down, the text in the first column says “Multiply the inner terms.” The middle column contains the product of y minus 7 and y plus 4 again, with a third arrow extending from the minus 7 in the first binomial to the y in the second binomial. The third column contains y squared plus 4y minus 7y plus blank. One row down, the text in the first column says “Multiply the last terms.” The second column contains the product of y minus 7 and y plus 4 again, with a fourth arrow extending from minus 7 in the first binomial to 4 in the second binomial. In the third column is the full expression, y squared plus 4y minus 7y minus 28, with each letter of FOIL beneath each of the terms. At the bottom of the image, the text in the first column says “Combine like terms.” In the right column is y squared minus 3y minus 28.

Try It 6.77

Multiply: $(x - 7) (x + 5) . (x - 7) (x + 5) .$

Try It 6.78

Multiply: $(b - 3) (b + 6) . (b - 3) (b + 6) .$

Example 6.40

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

Show solution

Solution

This figure has three columns. At the top of the figure, the second column contains the product of two binomials, 4x plus 3 and 2x minus 5. One row down, the text in the first column says “Multiply the first terms. 4x times 2x.” The second column contains 8x squared plus blank plus blank plus blank. Beneath 8x squared is the letter F and beneath each blank are the letters O, I, and L, respectively. One row down, the text in the first column says “Multiply the outer terms. 4x times negative 5.” The second column contains 8x squared minus 20x plus blank plus blank. Below 8x squared is F, below 20x is O, and below the blanks are I and L. One row down, the text in the first column says “Multiply the inner terms. 3 times 2x.” The second column contains 8x squared minus 20x plus 6x plus blank. One row down, the text in the first column says “Multiply the last terms. 3 times negative 5.” The second column contains the full expression, 8x squared minus 20x plus 6x minus 15, with each letter of FOIL beneath each of the terms. At the bottom of the image, the text in the first column says “Combine like terms.” In the right column is 8x squared minus 14x minus 15. In the third column is the product of the two binomials again, 4x plus 3 times 2x minus 5. An arrow extends from 4x in the first binomial to 2x in the second binomial. A second arrow extends from 4x in the first binomial to minus 5 in the second binomial. A third arrow extends from 3 in the first binomial to 2x in the second binomial. A fourth arrow extends from 3 in the first binomial to minus 5 in the second binomial.

Try It 6.79

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

Try It 6.80

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

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 6.41

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

Show solution

Solution



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

Try It 6.81

Multiply: $(10 c - d) (c - 6) . (10 c - d) (c - 6) .$

Try It 6.82

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

Be careful of the exponents in the next example.

Example 6.42

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

Show solution

Solution



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

Try It 6.83

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

Try It 6.84

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

Example 6.43

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

Show solution

Solution


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

Try It 6.85

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

Try It 6.86

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

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.

Example 6.44

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

Show solution

Solution

It does not matter which binomial goes on the top.

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

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

This figure has two columns. In the left column is the product of two binomials, 3y minus 1 and 2y minus 6. Below this is 6y squared minus 2y minus 18y plus 6. Below this is 6y squared minus 20y plus 6. In the right column is the vertical multiplication of 3y minus 1 and 2y minus 6. Below this is the partial product negative 18y plus 6. Below this is the partial product 6y squared minus 2y. Below this is 6y squared minus 20y plus 6.

Try It 6.87

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

Try It 6.88

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

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.

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 6.45

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

Show solution

Solution


Distribute.
Multiply.
Combine like terms.

Try It 6.89

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

Try It 6.90

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

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

Example 6.46

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

Show solution

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.

Try It 6.91

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

Try It 6.92

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

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.

Multiplying a Trinomial by a Binomial

To multiply a trinomial by a binomial, use the:

Distributive Property
Vertical Method

Media

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

Practice Makes Perfect

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

173.

$4 (w + 10) 4 (w + 10)$

174.

$6 (b + 8) 6 (b + 8)$

175.

$−3 (a + 7) −3 (a + 7)$

176.

$−5 (p + 9) −5 (p + 9)$

177.

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

178.

$7 (y - 4) 7 (y - 4)$

179.

$−3 (k - 4) −3 (k - 4)$

180.

$−8 (j - 5) −8 (j - 5)$

181.

$q (q + 5) q (q + 5)$

182.

$k (k + 7) k (k + 7)$

183.

$− b (b + 9) − b (b + 9)$

184.

$− y (y + 3) − y (y + 3)$

185.

$− x (x - 10) − x (x - 10)$

186.

$− p (p - 15) − p (p - 15)$

187.

$6 r (4 r + s) 6 r (4 r + s)$

188.

$5 c (9 c + d) 5 c (9 c + d)$

189.

$12 x (x - 10) 12 x (x - 10)$

190.

$9 m (m - 11) 9 m (m - 11)$

191.

$−9 a (3 a + 5) −9 a (3 a + 5)$

192.

$−4 p (2 p + 7) −4 p (2 p + 7)$

193.

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

194.

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

195.

$−8 x (x^{2} + 2 x - 15) −8 x (x^{2} + 2 x - 15)$

196.

$−5 t (t^{2} + 3 t - 18) −5 t (t^{2} + 3 t - 18)$

197.

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

198.

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

199.

$−8 y (y^{2} + 2 y - 15) −8 y (y^{2} + 2 y - 15)$

200.

$−5 m (m^{2} + 3 m - 18) −5 m (m^{2} + 3 m - 18)$

201.

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

202.

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

203.

$−4 z^{2} (3 z^{2} + 12 z - 1) −4 z^{2} (3 z^{2} + 12 z - 1)$

204.

$−3 x^{2} (7 x^{2} + 10 x - 1) −3 x^{2} (7 x^{2} + 10 x - 1)$

205.

$(2 m - 9) m (2 m - 9) m$

206.

$(8 j - 1) j (8 j - 1) j$

207.

$(w - 6) \cdot 8 (w - 6) \cdot 8$

208.

$(k - 4) \cdot 5 (k - 4) \cdot 5$

209.

$4 (x + 10) 4 (x + 10)$

210.

$6 (y + 8) 6 (y + 8)$

211.

$15 (r - 24) 15 (r - 24)$

212.

$12 (v - 30) 12 (v - 30)$

213.

$−3 (m + 11) −3 (m + 11)$

214.

$−4 (p + 15) −4 (p + 15)$

215.

$−8 (z - 5) −8 (z - 5)$

216.

$−3 (x - 9) −3 (x - 9)$

217.

$u (u + 5) u (u + 5)$

218.

$q (q + 7) q (q + 7)$

219.

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

220.

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

221.

$6 x (4 x + y) 6 x (4 x + y)$

222.

$5 a (9 a + b) 5 a (9 a + b)$

223.

$5 p (11 p - 5 q) 5 p (11 p - 5 q)$

224.

$12 u (3 u - 4 v) 12 u (3 u - 4 v)$

225.

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

226.

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

227.

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

228.

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

229.

$−8 y (y^{2} + 2 y - 15) −8 y (y^{2} + 2 y - 15)$

230.

$−5 m (m^{2} + 3 m - 18) −5 m (m^{2} + 3 m - 18)$

231.

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

232.

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

233.

$−4 z^{2} (3 z^{2} + 12 z - 1) −4 z^{2} (3 z^{2} + 12 z - 1)$

234.

$−3 x^{2} (7 x^{2} + 10 x - 1) −3 x^{2} (7 x^{2} + 10 x - 1)$

235.

$(2 y - 9) y (2 y - 9) y$

236.

$(8 b - 1) b (8 b - 1) b$

Multiply a Binomial by a Binomial

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

237.

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

238.

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

239.

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

240.

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

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

241.

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

242.

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

243.

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

244.

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

245.

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

246.

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

247.

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

248.

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

249.

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

250.

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

251.

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

252.

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

253.

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

254.

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

255.

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

256.

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

257.

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

258.

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

259.

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

260.

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

261.

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

262.

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

263.

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

264.

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

265.

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

266.

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

Multiply a Trinomial by a Binomial

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

267.

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

268.

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

269.

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

270.

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

In the following exercises, multiply. Use either method.

271.

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

272.

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

273.

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

274.

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

Mixed Practice

275.

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

276.

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

277.

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

278.

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

279.

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

280.

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

281.

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

282.

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

283.

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

284.

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

285.

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

286.

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

287.

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

288.

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

289.

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

290.

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

291.

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

292.

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

293.

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

Everyday Math

294.

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 $10 + 3 10 + 3$ and 15 as $10 + 5 10 + 5$ .

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

295.

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 $20 - 2 20 - 2$ and 17 as $20 - 3 20 - 3$ .

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

Writing Exercises

296.

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

297.

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

298.

Multiply the following:

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

Explain the pattern that you see in your answers.

299.

Multiply the following:

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

Explain the pattern that you see in your answers.

300.

Multiply the following:

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

Explain the pattern that you see in your answers.

301.

Multiply the following:

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

Explain the pattern that you see in your answers.

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?