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A vertical multiplication problem is shown. 23 times 46 is written with a line underneath. Beneath the line is 138. Beside 138 is written “partial product.” Beneath 138 is 92. Beside 92 is written “partial product.” Beneath 92 is a line and 1058. Beside 1058 is written “product.”

You start by multiplying 23 by 6 to get 138 .

Then you 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.

Multiply using the vertical method: ( 5 x 1 ) ( 2 x 7 ) .

Solution

It does not matter which binomial goes on the top. Line up the columns when you multiply as we did when we multiplied 23 ( 46 ) .

.
Multiply 2 x 7 by −1 . .
Multiply 2 x 7 by 5 x . .
Add like terms. .

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

On the left, 5x minus 1 times 2x minus 7 is shown. Below that is 10 x squared minus 35x minus 2x plus 7. The first two terms are in blue, the second two in red. Beneath that is 10 x squared minus 37x plus 7. On the right, a vertical multiplication problem is shown. 2xx minus 7 times 5x minus 1 is written with a line underneath. Beneath the line is a red negative 2x plus 7. Beneath that is 10 x squared minus 35 x in blue. Beneath that, there is another line. Beneath that line is 10 x squared minus 37x plus 7.

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Multiply using the vertical method: ( 4 m 9 ) ( 3 m 7 ) .

12 m 2 − 55 m + 63

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Multiply using the vertical method: ( 6 n 5 ) ( 7 n 2 ) .

42 n 2 − 47 n + 10

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We have now used three methods for multiplying binomials. Be sure to practice each method, and try to decide which one you prefer. The three methods are listed here 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, the FOIL method 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: ( x + 3 ) ( 2 x 2 5 x + 8 ) .

Solution

.
Distribute. .
Multiply. 2 x 3 5 x 2 + 8 x + 6 x 2 15 x + 24
Combine like terms. 2 x 3 + x 2 7 x + 24
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Multiply using the Distributive Property: ( y 1 ) ( y 2 7 y + 2 ) .

y 3 − 8 y 2 + 9 y − 2

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

3 x 3 + 2 x 2 − 3 x + 10

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

Multiply using the Vertical Method: ( x + 3 ) ( 2 x 2 5 x + 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 x 2 5 x + 8 ) by 3. .
Multiply ( 2 x 2 5 x + 8 ) by x . .
Add like terms. .
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Multiply using the Vertical Method: ( y 1 ) ( y 2 7 y + 2 ) .

y 3 − 8 y 2 + 9 y − 2

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

3 x 3 + 2 x 2 − 3 x + 10

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Key concepts

  • Use the FOIL method for multiplying two binomials.
    Step 1. Multiply the First terms. .
    Step 2. Multiply the Outer terms.
    Step 3. Multiply the Inner terms.
    Step 4. Multiply the Last terms.
    Step 5. Combine like terms, when possible.
  • Multiplying Two Binomials: To multiply binomials, use the:
    • Distributive Property
    • FOIL Method
    • Vertical Method
  • Multiplying a Trinomial by a Binomial: To multiply a trinomial by a binomial, use the:
    • Distributive Property
    • Vertical Method

Practice makes perfect

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

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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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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6 x ( 4 x + y )

24 x 2 + 6 xy

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5 p ( 11 p 5 q )

55 p 2 − 25 pq

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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

2 y 2 − 9 y

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( w 6 ) · 8

8 w − 48

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Multiply a Binomial by a Binomial

In the following exercises, multiply the following binomials using different methods.

( x + 4 ) ( x + 6 )

x 2 + 10 x + 24

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( n + 12 ) ( n 3 )

n 2 + 9 n − 36

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( y + 8 ) ( y + 3 )

y 2 + 11 y + 24

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( a + 6 ) ( a + 16 )

a 2 + 22 a + 96

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( u 5 ) ( u 9 )

u 2 − 14 u + 45

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( z 10 ) ( z 22 )

z 2 − 32 z + 220

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( b 5 ) ( b 24 )

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( x 4 ) ( x + 7 )

x 2 + 3 x − 28

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( v + 12 ) ( v 5 )

v 2 + 7 v − 60

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( 6 n + 5 ) ( n + 1 )

6 n 2 + 11 n + 5

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( 2 m 9 ) ( 10 m + 1 )

20 m 2 − 88 m − 9

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( 5 r 4 ) ( 12 r + 1 )

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( 4 c 1 ) ( 4 c + 1 )

16 c 2 − 1

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( 8 n 1 ) ( 8 n + 1 )

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( 3 u 8 ) ( 5 u 14 )

15 u 2 − 82 u + 112

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( 2 q 5 ) ( 7 q 11 )

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

2 a 2 + 5 ab + 3 b 2

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( 5 x y ) ( x 4 )

5 x 2 − 20 x xy + 4 y

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( 4 z y ) ( z 6 )

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Multiply a Trinomial by a Binomial

In the following exercises, using either method.

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

u 3 + 7 u 2 + 14 u + 8

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( x + 5 ) ( x 2 + 8 x + 3 )

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

3 a 3 + 31 a 2 + 5 a − 50

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( n + 8 ) ( 4 n 2 + n 7 )

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( y 6 ) ( y 2 10 y + 9 )

y 3 − 16 y 2 + 69 y − 54

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( k 3 ) ( k 2 8 k + 7 )

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( 2 x + 1 ) ( x 2 5 x 6 )

2 x 3 − 9 x 2 − 17 x − 6

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( 5 v + 1 ) ( v 2 6 v 10 )

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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?
  1. 195
  2. 195
  3. Answers will vary.
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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?
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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?

Answers will vary.

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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?

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Self check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

.

What does this checklist tell you about your mastery of this section? What steps will you take to improve?

Questions & Answers

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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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