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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

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.
Dojzae Reply
LeBron needs 150 milliliters of a 30% solution of sulfuric acid for a lab experiment but only has access to a 25% and a 50% solution. How much of the 25% and how much of the 50% solution should he mix to make the 30% solution?
Xona Reply
5%
Michael
hey everyone how to do algebra
The Reply
Felecia answer 1.5 hours before he reaches her
Adriana Reply
I would like to solve the problem -6/2x
rachel Reply
12x
Andrew
how
Christian
Does the x represent a number or does it need to be graphed ?
latonya
-3/x
Venugopal
-3x is correct
Atul
Arnold invested $64,000, some at 5.5% interest and the rest at 9%. How much did he invest at each rate if he received $4,500 in interest in one year?
Stephanie Reply
Tickets for the community fair cost $12 for adults and $5 for children. On the first day of the fair, 312 tickets were sold for a total of $2204. How many adult tickets and how many child tickets were sold?
Alpha Reply
220
gayla
Three-fourths of the people at a concert are children. If there are 87 children, what is the total number of people at the concert?
Tsimmuaj Reply
Erica earned a total of $50,450 last year from her two jobs. The amount she earned from her job at the store was $1,250 more than four times the amount she earned from her job at the college. How much did she earn from her job at the college?
Tsimmuaj
Erica earned a total of $50,450 last year from her two jobs. The amount she earned from her job at the store was $1,250 more than four times the amount she earned from her job at the college. How much did she earn from her job at the college?
Tsimmuaj
? Is there anything wrong with this passage I found the total sum for 2 jobs, but found why elaborate on extra If I total one week from the store *4 would = the month than the total is = x than x can't calculate 10 month of a year
candido
what would be wong
candido
87 divided by 3 then multiply that by 4. 116 people total.
Melissa
the actual number that has 3 out of 4 of a whole pie
candido
was having a hard time finding
Teddy
use Matrices for the 2nd question
Daniel
One number is 11 less than the other number. If their sum is increased by 8, the result is 71. Find the numbers.
Tsimmuaj Reply
26 + 37 = 63 + 8 = 71
gayla
26+37=63+8=71
ziad
11+52=63+8=71
Thisha
how do we know the answer is correct?
Thisha
23 is 11 less than 37. 23+37=63. 63+8=71. that is what the question asked for.
gayla
23 +11 = 37. 23+37=63 63+8=71
Gayla
by following the question. one number is 11 less than the other number 26+11=37 so 26+37=63+8=71
Gayla
your answer did not fit the guidelines of the question 11 is 41 less than 52.
gayla
71-8-11 =52 is this correct?
Ruel
let the number is 'x' and the other number is "x-11". if their sum is increased means: x+(x-11)+8 result will be 71. so x+(x-11)+8=71 2x-11+8=71 2x-3=71 2x=71+3 2x=74 1/2(2x=74)1/2 x=37 final answer
tesfu
just new
Muwanga
Amara currently sells televisions for company A at a salary of $17,000 plus a $100 commission for each television she sells. Company B offers her a position with a salary of $29,000 plus a $20 commission for each television she sells. How televisions would Amara need to sell for the options to be equal?
Tsimmuaj Reply
yes math
Kenneth
company A 13 company b 5. A 17,000+13×100=29,100 B 29,000+5×20=29,100
gayla
need help with math to do tsi test
Toocute
me too
Christian
have you tried the TSI practice test ***tsipracticetest.com
gayla
DaMarcus and Fabian live 23 miles apart and play soccer at a park between their homes. DaMarcus rode his bike for 34 of an hour and Fabian rode his bike for 12 of an hour to get to the park. Fabian’s speed was 6 miles per hour faster than DaMarcus’s speed. Find the speed of both soccer players.
gustavo Reply
?
Ann
DaMarcus: 16 mi/hr Fabian: 22 mi/hr
Sherman
Joy is preparing 20 liters of a 25% saline solution. She has only a 40% solution and a 10% solution in her lab. How many liters of the 40% solution and how many liters of the 10% solution should she mix to make the 25% solution?
Wenda Reply
15 and 5
32 is 40% , & 8 is 10 % , & any 4 letters is 5%.
Karen
It felt that something is missing on the question like: 40% of what solution? 10% of what solution?
Jhea
its confusing
Sparcast
3% & 2% to complete the 25%
Sparcast
because she already has 20 liters.
Sparcast
ok I was a little confused I agree 15% & 5%
Sparcast
8,2
Karen
Jim and Debbie earned $7200. Debbie earned $1600 more than Jim earned. How much did they earned
Arleathia Reply
5600
Gloria
1600
Gloria
Bebbie: 4,400 Jim: 2,800
Jhea
A river cruise boat sailed 80 miles down the Mississippi River for 4 hours. It took 5 hours to return. Find the rate of the cruise boat in still water and the rate of the current.
Sunnyshay Reply
A veterinarian is enclosing a rectangular outdoor running area against his building for the dogs he cares for. He needs to maximize the area using 100 feet of fencing. The quadratic equation A=x(100−2x) gives the area, A , of the dog run for the length, x , of the building that will border the dog run. Find the length of the building that should border the dog run to give the maximum area, and then find the maximum area of the dog run.
Elizabeth Reply
ggfcc
Mike

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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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