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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

hello, I have algebra phobia. Subtracting negative numbers always seem to get me confused.
Alicia Reply
what do you need help in?
Felix
subtracting a negative....is adding!!
Heather
look at the numbers if they have different signs, it's like subtracting....but you keep the sign of the largest number...
Felix
for example.... -19 + 7.... different signs...subtract.... 12 keep the sign of the "largest" number 19 is bigger than 7.... 19 has the negative sign... Therefore, -12 is your answer...
Felix
—12
Niazmohammad
Thanks Felix.l also get confused with signs.
Esther
Thank you for this
Shatey
ty
Graham
Bruce drives his car for his job. The equation R=0.575m+42 models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day. Find the amount Bruce is reimbursed on a day when he drives 220 miles
John Reply
168.50=R
Heather
john is 5years older than wanjiru.the sum of their years is27years.what is the age of each
achol Reply
46
mustee
j 17 w 11
Joseph
john is 16. wanjiru is 11.
Felix
27-5=22 22÷2=11 11+5=16
Joyce
where's the answers?
Ed Reply
I don't see where the answers are.
Ed
Cindy and Richard leave their dorm in Charleston at the same time. Cindy rides her bicycle north at a speed of 18 miles per hour. Richard rides his bicycle south at a speed of 14 miles per hour. How long will it take them to be 96 miles apart?
Maddy Reply
3
Christopher
18t+14t=96 32t=96 32/96 3
Christopher
show that a^n-b^2n is divisible by a-b
Florence Reply
What does 3 times your weight right now
Cherokee Reply
Use algebra to combine 39×5 and the half sum of travel of 59+30
Cherokee
What is the segment of 13? Explain
Cherokee
my weight is 49. So 3 times is 147
Cherokee
kg to lbs you goin to convert 2.2 or one if the same unit your going to time your body weight by 3. example if my body weight is 210lb. what would be my weight if I was 3 times as much in kg. that's you do 210 x3 = 630lb. then 630 x 2.2= .... hope this helps
tyler
How to convert grams to pounds?
paul
What is the lcm of 340
Kendra Reply
Yes
Cherokee
How many numbers each equal to y must be taken to make 15xy
Malik Reply
15x
Martin
15x
Asamoah
15x
Hugo
1y
Tom
1y x 15y
Tom
find the equation whose roots are 1 and 2
Adda Reply
(x - 2)(x -1)=0 so equation is x^2-x+2=0
Ranu
I believe it's x^2-3x+2
NerdNamedGerg
because the X's multiply by the -2 and the -1 and than combine like terms
NerdNamedGerg
find the equation whose roots are -1 and 4
Adda
Ans = ×^2-3×+2
Gee
find the equation whose roots are -2 and -1
Adda
(×+1)(×-4) = x^2-3×-4
Gee
yeah
Asamoah
Quadratic equations involving factorization
Winner Reply
there's a chatting option in the app wow
Nana
That's cool cool
Nana
Nice to meet you all
Nana
you too.
Joan
😃
Nana
Hey you all there are several Free Apps that can really help you to better solve type Equations.
Debra
Debra, which apps specifically. ..?
Nana
am having a course in elementary algebra ,any recommendations ?
samuel
Samuel Addai, me too at ucc elementary algebra as part of my core subjects in science
Nana
me too as part of my core subjects in R M E
Ken
at ABETIFI COLLEGE OF EDUCATION
Ken
ok great. Good to know.
Joan
5x + 1/3= 2x + 1/2
sanam
Plz solve this
sanam
5x - 3x = 1/2 - 1/3 2x = 1/6 x = 1/12
Ranu
Thks ranu
sanam
Please help me solve this 10x+14=-2x+38
Erica
the previous equation should be 3x = 1/6 x=1/18
Sriram
for the new one 10x + 2x = 38 - 14
Sriram
12x = 24 x=2
Sriram
10x + 14 = -2x +38 10x + 2x = 38 - 14 12x = 24 divide both sides by the coefficient of x, which is 12 therefore × = 2
vida
a trader gains 20 rupees loses 42 rupees and then gains ten rupees Express algebraically the result of his transactions
vinaya Reply
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
Kim is making eight gallons of punch from fruit juice and soda. The fruit juice costs $6.04 per gallon and the soda costs $4.28 per gallon. How much fruit juice and how much soda should she use so that the punch costs $5.71 per gallon?
Mohamed Reply
(a+b)(p+q+r)(b+c)(p+q+r)(c+a) (p+q+r)
muhammad Reply
really
Asamoah
4x-7y=8 2x-7y=1 what is the answer?
Ramil Reply
x=7/2 & y=6/7
Pbp
x=7/2 & y=6/7 use Elimination
Debra
true
bismark
factoriz e
usman
4x-7y=8 X=7/4y+2 and 2x-7y=1 x=7/2y+1/2
Peggie
Ok cool answer peggie
Frank
thanks
Ramil
x=7/2 y=6/7
Asamoah
Thanks , all of you are correct.
Joseph

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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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