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Section 5.3

Special Products

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Objective #1 Use FOIL in polynomial multiplication.

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Special Products

In this section we will use the distributive property to develop patterns that will allow us to multiply some special binomials quickly.

We will find the product of two binomials using a method called FOIL.

We will learn a formula for finding the square of a binomial sum. We will also learn formula for finding the product of the sum and difference of two terms.

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Multiplying Polynomials - FOIL

dbcxbdaxcxaxdcxbax

Using the FOIL Method to Multiply Binomials

first

outside

inside

last F O I L

Product of First terms

Product of Outside terms

Product of Inside terms

Product of Last terms

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Multiplying Polynomials - FOIL

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

Multiply . 1534 xx

1534 xx 13531454 xxxx

Combine like terms

Multiply315420 2 xxx

31920 2 xx

F O I Lfirst

outside

inside

last

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Multiply: (5x + 2)(x + 7)

(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7

=5x2 + 35x + 2x +14=5x2 + 37x +14

Product of the first

terms

Product of the

outside terms

Product of the inside terms

Product of the last

terms

F O I L

FOIL Method

EXAMPLEEXAMPLE

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Multiply: (5x + 2)(x + 7)

(5x + 2)(x + 7) = 5x·x + 5x·7 + 2·x + 2·7

=5x2 + 35x + 2x +14=5x2 + 37x +14

Product of the first

terms

Product of the

outside terms

Product of the inside terms

Product of the last

terms

F O I L

FOIL Method

EXAMPLEEXAMPLE

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Objective #1: Example

1a. Multiply: ( 5)( 6)x x

OF I L

2

2

( 5)( 6) 6 5 5 6

6 5 30

11 30

x x x x x x

x x x

x x

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Objective #1: Example

1a. Multiply: ( 5)( 6)x x

OF I L

2

2

( 5)( 6) 6 5 5 6

6 5 30

11 30

x x x x x x

x x x

x x

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Objective #1: Example

1b. Multiply: (7 5)(4 3)x x

OF I L

2

2

(7 5)(4 3) 7 4 7 ( 3) 5 4 5( 3)

28 21 20 15

28 15

x x x x x x

x x x

x x

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Objective #1: Example

1b. Multiply: (7 5)(4 3)x x

OF I L

2

2

(7 5)(4 3) 7 4 7 ( 3) 5 4 5( 3)

28 21 20 15

28 15

x x x x x x

x x x

x x

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Objective #2

Multiply the sum and difference of two terms.

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(A + B)(A – B) = A2 – B2

The product of the sum and the difference of the same two terms is the square of the first term minus the square of the second term.

Multiplying the Sum and Difference of Two Terms

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2a. Multiply: (7 8)(7 8)y y

Since this product is of the form ( )( )A B A B ,

use the special–product formula 2 2( )( )A B A B A B .

first term second termsquared squared

2 2

2

(7 8)(7 8) (7 ) 8

49 64

y y y

y

Objective #2: Example

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2a. Multiply: (7 8)(7 8)y y

Since this product is of the form ( )( )A B A B ,

use the special–product formula 2 2( )( )A B A B A B .

first term second termsquared squared

2 2

2

(7 8)(7 8) (7 ) 8

49 64

y y y

y

Objective #2: Example

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Objective #2: Example

2b. Multiply: 3 3(2 3)(2 3)a a

Since this product is of the form ( )( )A B A B ,

use the special–product formula 2 2( )( )A B A B A B .

first term second termsquared squared

3 3 3 2 2

6

(2 3)(2 3) (2 ) 3

4 9

a a a

a

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Objective #2: Example

2b. Multiply: 3 3(2 3)(2 3)a a

Since this product is of the form ( )( )A B A B ,

use the special–product formula 2 2( )( )A B A B A B .

first term second termsquared squared

3 3 3 2 2

6

(2 3)(2 3) (2 ) 3

4 9

a a a

a

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Objective #3 Find the square of a binomial sum.

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(A + B)2 = A2 + 2AB + B2

The square of a binomial sum is the first term squared plus two times the product of the terms plus the last term squared.

The Square of a Binomial Sum

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Multiplying Polynomials – Special Formulas

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

Multiply .4 2yx

24 yx

222 2 BABABA

Use the special-product formula shown.

+ + = Product

+ +

2

Term

First

Terms theof

Product22

Term

Last

24x yx422y 22 816 yxyx

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Objective #3: Example

3a. Multiply: 2( 10)x

Use the special-product formula 2 2 2( ) 2 .A B A AB B

first term last term2 productsquared squaredof the terms

2 2 2

2

( 10) 2 10 10

20 100

x x x

x x

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Objective #3: Example

3a. Multiply: 2( 10)x

Use the special-product formula 2 2 2( ) 2 .A B A AB B

first term last term2 productsquared squaredof the terms

2 2 2

2

( 10) 2 10 10

20 100

x x x

x x

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Objective #3: Example

3b. Multiply: 2(5 4)x

Use the special-product formula 2 2 2( ) 2 .A B A AB B

first term last term2 productsquared squaredof the terms

2 2 2

2

(5 4) (5 ) 2 20 4

25 40 16

x x x

x x

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Objective #3: Example

3b. Multiply: 2(5 4)x

Use the special-product formula 2 2 2( ) 2 .A B A AB B

first term last term2 productsquared squaredof the terms

2 2 2

2

(5 4) (5 ) 2 20 4

25 40 16

x x x

x x

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Objective #4 Find the square of a binomial difference.

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(A – B)2 = A2 – 2AB + B2

The square of a binomial difference is the first term squared minus two times the product of the terms plus the last term squared.

The Square of a Binomial Difference

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Multiplying Polynomials – Special Formulas

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

Multiply . 43 2yx

243 yx

222 2 BABABA

Use the special-product formula shown.

– + = Product

– +

2

Term

First

Terms theof

Product22

Term

Last

23x yx 432 24y22 16249 yxyx

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4a. Multiply: 2( 9)x

Use the special-product formula 2 2 2( ) 2 .A B A AB B

first term last term2 productsquared squaredof the terms

2 2 2

2

( 9) 2 9 9

18 81

x x x

x x

Objective #4: Example

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4a. Multiply: 2( 9)x

Use the special-product formula 2 2 2( ) 2 .A B A AB B

first term last term2 productsquared squaredof the terms

2 2 2

2

( 9) 2 9 9

18 81

x x x

x x

Objective #4: Example

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Objective #4: Example

4b. Multiply: 2(7 3)x

Use the special-product formula 2 2 2( ) 2 .A B A AB B

first term last term2 productsquared squaredof the terms

2 2 2

2

(7 3) (7 ) 2 21 3

49 42 9

x x x

x x

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Objective #4: Example

4b. Multiply: 2(7 3)x

Use the special-product formula 2 2 2( ) 2 .A B A AB B

first term last term2 productsquared squaredof the terms

2 2 2

2

(7 3) (7 ) 2 21 3

49 42 9

x x x

x x

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Multiplying Polynomials – Special Formulas

The Square of a Binomial Sum

22 BABABA

The Square of a Binomial Difference

222 2 BABABA

222 2 BABABA

The Product of the Sum and Difference of Two Terms