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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

The product of a binomial, b plus 3, and a trinomial, 2 b squared minus 5 b plus 8. Two arrows extend from the trinomial, terminating at b and 3 in the binomial.
Distribute. The sum of two products, the product of b and 2 b squared minus 5 b plus 8, and the product of 3 and 2 b squared minus 5 b plus 8.
Multiply. 2 b cubed minus 5 b squared plus 8 b plus 6 b squared minus 15 b plus 24.
Combine like terms. 2 b cubed plus b squared minus 7 b plus 24.

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Multiply using the Distributive Property: ( y 3 ) ( y 2 5 y + 2 ) .

y 3 8 y 2 + 17 y 6

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Multiply using the Distributive Property: ( x + 4 ) ( 2 x 2 3 x + 5 ) .

2 x 3 + 5 x 2 7 x + 20

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

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Multiply using the Vertical Method: ( y 3 ) ( y 2 5 y + 2 ) .

y 3 8 y 2 + 17 y 6

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Multiply using the Vertical Method: ( x + 4 ) ( 2 x 2 3 x + 5 ) .

2 x 3 + 5 x 2 7 x + 20

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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

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

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:
  • Multiplying a Trinomial by a Binomial —To multiply a trinomial by a binomial, use the:
    • Distributive Property ( [link] )
    • Vertical Method ( [link] )

Practice makes perfect

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

−3 ( a + 7 )

−3 a 21

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2 ( x 7 )

2 x 14

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−3 ( k 4 )

−3 k + 12

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b ( b + 9 )

b 2 9 b

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x ( x 10 )

x 2 + 10 x

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6 r ( 4 r + s )

24 r 2 + 6 r s

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12 x ( x 10 )

12 x 2 120 x

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−9 a ( 3 a + 5 )

−27 a 2 45 a

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3 ( p 2 + 10 p + 25 )

3 p 2 + 30 p + 75

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−8 x ( x 2 + 2 x 15 )

−8 x 3 16 x 2 + 120 x

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−5 t ( t 2 + 3 t 18 )

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5 q 3 ( q 3 2 q + 6 )

5 q 6 10 q 4 + 30 q 3

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4 x 3 ( x 4 3 x + 7 )

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−8 y ( y 2 + 2 y 15 )

−8 y 3 16 y 2 + 120 y

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−5 m ( m 2 + 3 m 18 )

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5 q 3 ( q 2 2 q + 6 )

5 q 5 10 q 4 + 30 q 3

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9 r 3 ( r 2 3 r + 5 )

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−4 z 2 ( 3 z 2 + 12 z 1 )

−12 z 4 48 z 3 + 4 z 2

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−3 x 2 ( 7 x 2 + 10 x 1 )

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( 2 m 9 ) m

2 m 2 9 m

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( w 6 ) · 8

8 w 48

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15 ( r 24 )

15 r 360

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−3 ( m + 11 )

−3 m 33

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−8 ( z 5 )

−8 z + 40

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n ( n 2 3 n )

n 3 3 n 2

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6 x ( 4 x + y )

24 x 2 + 6 x y

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5 p ( 11 p 5 q )

55 p 2 25 p q

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3 ( v 2 + 10 v + 25 )

3 v 2 + 30 v + 75

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2 n ( 4 n 2 4 n + 1 )

8 n 3 8 n 2 + 2 n

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3 r ( 2 r 2 6 r + 2 )

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−8 y ( y 2 + 2 y 15 )

−8 y 3 16 y 2 + 120 y

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−5 m ( m 2 + 3 m 18 )

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5 q 3 ( q 2 2 q + 6 )

5 q 5 10 q 4 + 30 q 3

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9 r 3 ( r 2 3 r + 5 )

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−4 z 2 ( 3 z 2 + 12 z 1 )

−12 z 4 48 z 3 + 4 z 2

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−3 x 2 ( 7 x 2 + 10 x 1 )

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( 2 y 9 ) y

18 y 2 9 y

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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 )

w 2 + 12 w + 35

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( p + 11 ) ( p 4 )

p 2 + 7 p 44

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

( x + 8 ) ( x + 3 )

x 2 + 11 x + 24

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( y 6 ) ( y 2 )

y 2 8 y + 12

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( w 4 ) ( w + 7 )

w 2 + 3 w 28

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( p + 12 ) ( p 5 )

p 2 + 7 p 60

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( 6 p + 5 ) ( p + 1 )

6 p 2 + 11 p + 5

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( 2 t 9 ) ( 10 t + 1 )

20 t 2 88 t 9

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( 3 r 8 ) ( 11 r + 1 )

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( 5 x y ) ( 3 x 6 )

15 x 2 3 x y 30 x + 6 y

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( 10 a b ) ( 3 a 4 )

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

2 a 2 + 5 a b + 3 b 2

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( 4 z y ) ( z 6 )

4 z 2 24 z z y + 6 y

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( 5 x y ) ( x 4 )

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( x 2 + 3 ) ( x + 2 )

x 3 + 2 x 2 + 3 x + 6

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( x 2 + 8 ) ( x 2 5 )

x 4 + 3 x 2 40

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( y 2 7 ) ( y 2 4 )

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

10 a 2 b 2 + 13 a b 3

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

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

24 p 2 q 2 42 p q + 15

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( 3 r s 7 ) ( 3 r s 4 )

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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 )

x 3 + 9 x 2 + 23 x + 15

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( u + 4 ) ( u 2 + 3 u + 2 )

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( y + 8 ) ( 4 y 2 + y 7 )

4 y 3 + 33 y 2 + y 56

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( a + 10 ) ( 3 a 2 + a 5 )

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In the following exercises, multiply. Use either method.

( w 7 ) ( w 2 9 w + 10 )

w 3 16 w 2 + 73 w 70

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( p 4 ) ( p 2 6 p + 9 )

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( 3 q + 1 ) ( q 2 4 q 5 )

3 q 3 11 q 2 19 q 5

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( 6 r + 1 ) ( r 2 7 r 9 )

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Mixed Practice

( 10 y 6 ) + ( 4 y 7 )

14 y 13

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( 15 p 4 ) + ( 3 p 5 )

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( x 2 4 x 34 ) ( x 2 + 7 x 6 )

−11 x 28

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( j 2 8 j 27 ) ( j 2 + 2 j 12 )

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5 q ( 3 q 2 6 q + 11 )

15 q 3 30 q 2 + 55 q

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8 t ( 2 t 2 5 t + 6 )

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( s 7 ) ( s + 9 )

s 2 + 2 s 63

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( y 2 2 y ) ( y + 1 )

y 3 y 2 2 y

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( a 2 3 a ) ( 4 a + 5 )

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( 3 n 4 ) ( n 2 + n 7 )

3 n 3 n 2 25 n + 28

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( 6 k 1 ) ( k 2 + 2 k 4 )

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( 7 p + 10 ) ( 7 p 10 )

49 p 2 100

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( 3 y + 8 ) ( 3 y 8 )

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( 4 m 2 3 m 7 ) m 2

4 m 4 3 m 3 7 m 2

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( 15 c 2 4 c + 5 ) c 4

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

25 a 2 + 70 a b + 49 b 2

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( 3 x 11 y ) ( 3 x 11 y )

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( 4 y + 12 z ) ( 4 y 12 z )

16 y 2 144 z 2

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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 .

  1. Multiply ( 10 + 3 ) ( 10 + 5 ) by the FOIL method.
  2. Multiply 13 · 15 without using a calculator.
  3. Which way is easier for you? Why?
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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 .

  1. Multiply ( 20 2 ) ( 20 3 ) by the FOIL method.
  2. Multiply 18 · 17 without using a calculator.
  3. Which way is easier for you? Why?

306 306 Answers will vary.

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Writing exercises

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

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

Answers will vary.

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Multiply the following:

( x + 2 ) ( x 2 ) ( y + 7 ) ( y 7 ) ( w + 5 ) ( w 5 )

Explain the pattern that you see in your answers.

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Multiply the following:

( m 3 ) ( m + 3 ) ( n 10 ) ( n + 10 ) ( p 8 ) ( p + 8 )

Explain the pattern that you see in your answers.

Answers may vary.

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Multiply the following:

( p + 3 ) ( p + 3 ) ( q + 6 ) ( q + 6 ) ( r + 1 ) ( r + 1 )

Explain the pattern that you see in your answers.

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Multiply the following:

( x 4 ) ( x 4 ) ( y 1 ) ( y 1 ) ( z 7 ) ( z 7 )

Explain the pattern that you see in your answers.

Answers may vary.

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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?

Questions & Answers

Larry and Tom were standing next to each other in the backyard when Tom challenged Larry to guess how tall he was. Larry knew his own height is 6.5 feet and when they measured their shadows, Larry’s shadow was 8 feet and Tom’s was 7.75 feet long. What is Tom’s height?
genevieve Reply
6.25
Ciid
Wayne is hanging a string of lights 57 feet long around the three sides of his patio, which is adjacent to his house. the length of his patio, the side along the house, is 5 feet longer than twice it's width. Find the length and width of the patio.
Katherine Reply
complexed. could you please help me figure out?
Ciid
(sin=opp/adj) (tan= opp/adj) cos=hyp/adj
tyler
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tyler
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tyler
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tyler
SOH = Sine is Opposite over Hypotenuse. CAH= Cosine is Adjacent over Hypotenuse. TOA = Tangent is Opposite over Adjacent.
tyler
H=57 and O=285 figure out what the adjacent?
tyler
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 many televisions would Amara need to sell for the options to be equal?
Marisol Reply
what is the quantity and price of the televisions for both options?
karl
Amara has to sell 120 televisions to make 29,000 of the salary of company B. 120 * TV 100= commission 12,000+ 17,000 = of company a salary 29,000 17000+
Ciid
Amara has to sell 120 televisions to make 29,000 of the salary of company B. 120 * TV 100= commission 12,000+ 17,000 = of company a salary 29,000
Ciid
I'm mathematics teacher from highly recognized university.
Mzo Reply
here a question professor How many soldiers are there in a group of 27 sailors and soldiers if there are four fifths as many sailors as soldiers? can you write out the college you went to with the name of the school you teach at and let me know the answer I've got it to be honest with you
tyler
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Helpful Reply
Yes
Val
chapter 1 foundations 1.2 exercises variables and algebraic symbols
theresa Reply
June needs 45 gallons of punch for a party and has 2 different coolers to carry it in. The bigger cooler is 5 times as large as the smaller cooler. How many gallons can each cooler hold? Enter the answers in decimal form.
Samer Reply
Joseph would like to make 12 pounds of a coffee blend at a cost of $6.25 per pound. He blends Ground Chicory at $4.40 a pound with Jamaican Blue Mountain at $8.84 per pound. How much of each type of coffee should he use?
Samer
4x6.25= $25 coffee blend 4×4.40= $17.60 ground chicory 4x8.84= 35.36 blue mountain. In total they will spend for 12 pounds $77.96 they will spend in total
tyler
DaMarcus and Fabian live 23 miles apart and play soccer at a park between their homes. DaMarcus rode his bike for three-quarters of an hour and Fabian rode his bike for half an hour to get to the park. Fabian’s speed was six miles per hour faster than DaMarcus’ speed. Find the speed of both soccer players.
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Danteii
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Daniel
How many soldiers are there in a group of 27 sailors and soldiers if there are four fifths many sailors as soldiers?
tyler
What is the domain and range of heaviside
Christopher Reply
What is the domain and range of Heaviside and signum
Christopher
25-35
Fazal
The hypotenuse of a right triangle is 10cm long. One of the triangle’s legs is three times the length of the other leg. Find the lengths of the three sides of the triangle.
Edi Reply
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Mum Reply
A 50% antifreeze solution is to be mixed with a 90% antifreeze solution to get 200 liters of a 80% solution. How many liters of the 50% solution and how many liters of the 90% solution will be used?
Edi Reply
June needs 45 gallons of punch for a party and has 2 different coolers to carry it in. The bigger cooler is 5 times as large as the smaller cooler. How many gallons can each cooler hold?
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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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