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Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1
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