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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

Before you get started, take this readiness quiz.

  1. Distribute: 2 ( x + 3 ) .
    If you missed this problem, review [link] .
  2. Combine like terms: x 2 + 9 x + 7 x + 63 .
    If you missed this problem, review [link] .

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

4 times x plus 3. Two arrows extend from 4, terminating at x and 3.
Distribute. 4 times x plus 4 times 3.
Simplify. 4 x plus 12.

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Multiply: 5 ( x + 7 ) .

5 x + 35

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Multiply: 3 ( y + 13 ) .

3 y + 39

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Multiply: y ( y 2 ) .

Solution

y times y minus 2. Two arrows extend from the coefficient y, terminating at the y and minus 2 in parentheses.
Distribute. y times y minus y times 2.
Simplify. y squared minus 2 y.

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Multiply: x ( x 7 ) .

x 2 7 x

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Multiply: d ( d 11 ) .

d 2 11 d

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Multiply: 7 x ( 2 x + y ) .

Solution

7 x times 2 x plus y. Two arrows extend from 7x, terminating at 2x and y.
Distribute. 7 x times 2 x plus 7 x times y.
Simplify. 14 x squared plus 7 x y.

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Multiply: 5 x ( x + 4 y ) .

5 x 2 + 20 x y

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Multiply: 2 p ( 6 p + r ) .

12 p 2 + 2 p r

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Multiply: −2 y ( 4 y 2 + 3 y 5 ) .

Solution

Negative 2 y times 4 y squared plus 3 y minus 5. Three arrows extend from negative 2 y, terminating at 4 y squared, 3 y, and minus 5.
Distribute. Negative 2 y times 4 y squared plus negative 2 y times 3 y minus negative 2 y times 5.
Simplify. Negative 8 y cubed minus 6 y squared plus 10 y.

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Multiply: −3 y ( 5 y 2 + 8 y 7 ) .

−15 y 3 24 y 2 + 21 y

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

8 x 4 24 x 3 + 20 x 2

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Multiply: 2 x 3 ( x 2 8 x + 1 ) .

Solution

2 x cubed times x squared minus 8 x plus 1. Three arrows extend from 2 x cubed, terminating at x squared, minus 8 x, and 1.
Distribute. 2 x cubed times x squared plus 2 x cubed times negative 8 x plus 2 x cubed times 1.
Simplify. 2 x to the fifth power minus 16 x to the fourth power plus 2 x cubed.

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

12 x 3 20 x 2 + 12 x

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Multiply: −6 a 3 ( 3 a 2 2 a + 6 ) .

−18 a 5 + 12 a 4 36 a 3

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Multiply: ( x + 3 ) p .

Solution

The monomial is the second factor. x plus 3, in parentheses, times p. Two arrows extend from the p, terminating at x and 3.
Distribute. x times p plus 3 times p.
Simplify. x p plus 3 p.

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Multiply: ( x + 8 ) p .

x p + 8 p

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Multiply: ( a + 4 ) p .

a p + 4 p

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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 [link] , 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: x p plus 3 p.
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. x squared plus 7 x plus 3 x plus 21.
Combine like terms. x squared plus 10 x plus 21.

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

The product of two binomials, y plus 5 and y plus 8. Two arrows extend from y plus 8, terminating at the y and the 5 in the first binomial.
Distribute ( y + 8). The sum of two products, the product of y and y plus 8, plus the product of 5 and y plus 8.
Distribute again y squared plus 8 y plus 5 y plus 40.
Combine like terms. y squared plus 13 y plus 40.

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

x 2 + 17 x + 72

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

20 x 2 + 51 x + 27

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

Solution

The product of two binomials, 2 y plus 5 and 3 y plus 4. Two arrows extend from 3y plus 4, terminating at 2y and 5 in the first binomial.
Distribute (3 y + 4). The sum of two products, the product of 2 y and 3 y plus 4, plus the product of 5 and 3 y plus 4.
Distribute again 6 y squared plus 8 y plus 15 y plus 20.
Combine like terms. 6 y squared plus 23 y plus 20.

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

12 b 2 + 38 b + 30

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

a 2 + 17 a + 70

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

Solution

The product of two binomials, 4y plus 3 and 2 y minus 5. Two arrows extend from 2y minus 5, terminating at 4 y and 3 in the first binomial.
Distribute. The sum of two products, the product of 4y and 2y minus 5, plus the product of 3 and 2y minus 5.
Distribute again. 8 y squared minus 20 y plus 6 y minus 15.
Combine like terms. 18 y squared minus 14 y minus 15.

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

30 y 2 3 y 6

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

15 c 2 + 14 c 8

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

Solution

The product of two binomials, x minus 2 and x minus y. Two arrows extend from x minus y, terminating at x and 2 in the first binomial.
Distribute. The difference of two products. The product of x and x minus 7, minus the product of 2 and x minus y.
Distribute again. x squared minus x y minus 2 x plus 2 y.
There are no like terms to combine.

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

a 2 a b + 7 a 7 b

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

x 2 x y + 5 x 5 y

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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 [link] , 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.

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

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 ‘ F irst, O uter, I nner, L ast’. The word FOIL is easy to remember and ensures we find all four products.

Questions & Answers

p-2/3=5/6 how do I solve it with explanation pls
Adedamola Reply
P=3/2
Vanarith
1/2p2-2/3p=5p/6
James
is y=7/5 a solution of 5y+3=10y-4
Adedamola Reply
yes
James
Lucinda has a pocketful of dimes and quarters with a value of $6.20. The number of dimes is 18 more than 3 times the number of quarters. How many dimes and how many quarters does Lucinda have?
Rhonda Reply
Find an equation for the line that passes through the point P ( 0 , − 4 ) and has a slope 8/9 .
Gabriel Reply
is that a negative 4 or positive 4?
Felix
y = mx + b
Felix
if negative -4, then -4=8/9(0) + b
Felix
-4=b
Felix
if positive 4, then 4=b
Felix
then plug in y=8/9x - 4 or y=8/9x+4
Felix
Macario is making 12 pounds of nut mixture with macadamia nuts and almonds. macadamia nuts cost $9 per pound and almonds cost $5.25 per pound. how many pounds of macadamia nuts and how many pounds of almonds should macario use for the mixture to cost $6.50 per pound to make?
Cherry Reply
Nga and Lauren bought a chest at a flea market for $50. They re-finished it and then added a 350 % mark - up
Makaila Reply
the sum of two Numbers is 19 and their difference is 15
Abdulai Reply
2, 17
Jose
interesting
saw
Felecia left her home to visit her daughter, driving 45mph. Her husband waited for the dog sitter to arrive and left home 20 minutes, or 13 hour later. He drove 55mph to catch up to Felecia. How long before he reaches her?
Rafi Reply
integer greater than 2 and less than 12
Emily Reply
2 < x < 12
Felix
I'm guessing you are doing inequalities...
Felix
Actually, translating words into algebraic expressions / equations...
Felix
hi
Darianna
hello
Mister
Eric here
Eric
He charges $125 per job. His monthly expenses are $1,600. How many jobs must he work in order to make a profit of at least $2,400?
Alicia Reply
at least 20
Ayla
what are the steps?
Alicia
6.4 jobs
Grahame
32
Grahame
1600+2400= total amount with expenses. 4000/125= number of jobs needed to make that min profit of 2400. answer is 32
Orlando
He must work 32 jobs to make a profit
POP
what is algebra
Azhar Reply
repeated addition and subtraction of the order of operations. i love algebra I'm obsessed.
Shemiah
hi
Krekar
Eric here. I'm a parent. 53 years old. I have never taken algebra. I want to learn.
Eric
One-fourth of the candies in a bag of M&M’s are red. If there are 23 red candies, how many candies are in the bag?
Leanna Reply
they are 92 candies in the bag
POP
rectangular field solutions
Navin Reply
What is this?
Donna
t
muqtaar
the proudact of 3x^3-5×^2+3 and 2x^2+5x-4 in z7[x]/ is
anas Reply
?
Choli
a rock is thrown directly upward with an initial velocity of 96feet per second from a cliff 190 feet above a beach. The hight of tha rock above the beach after t second is given by the equation h=_16t^2+96t+190
Usman
Stella bought a dinette set on sale for $725. The original price was $1,299. To the nearest tenth of a percent, what was the rate of discount?
Manhwa Reply
44.19%
Scott
40.22%
Terence
44.2%
Orlando
I don't know
Donna
if you want the discounted price subtract $725 from $1299. then divide the answer by $1299. you get 0.4419... but as percent you get 44.19... but to the nearest tenth... round .19 to .2 and you get 44.2%
Orlando
you could also just divide $725/$1299 and then subtract it from 1. then you get the same answer.
Orlando

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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