Algebra-Trigonometry-CynthiaYoung-3ed7.pdf - 8/3/12 11:16...

0.3
Polynomials: Basic Operations 31
EXAMPLE 5
Multiplying Two Polynomials
Multiply and simplify
= 2 x 2 ( x 2 - 5 x + 7) - 3 x ( x 2 - 5 x + 7) + 1( x 2 - 5 x + 7)
( x
2
- 5 x + 7) (2 x
2
- 3 x + 1) .
First
Inner
Product of
First Terms
Product of
Last Terms
Outer
Last
Product of
Outer Terms
Product of
Inner Terms
Special Products
The method outlined for multiplying polynomials works for all products of polynomials. For
the special case when both polynomials are binomials, the FOIL method can also be used.
W
ORDS
M
ATH
Apply the distributive property. (5 x 1)(2 x 3) 5 x (2 x 3) 1(2 x 3)
Apply the distributive property. 5 x (2 x ) 5 x (3) 1(2 x ) 1(3)
Multiply each set of monomials. 10 x 2 15 x 2 x 3
Combine like terms. 10 x
2
13 x 3
The FOIL method finds the
products of the F irst terms,
O uter terms, I nner terms, and (5 x 1)(2 x 3) 10 x 2 15 x 2 x 3
L ast terms.
EXAMPLE 6
Multiplying Binomials Using the FOIL Method
Multiply (3 x 1)(2 x 5) using the FOIL method.
Solution:
Multiply the first terms. (3 x )(2 x ) 6 x 2
Multiply the outer terms. (3 x )( 5) 15 x
Multiply the inner terms. (1)(2 x ) 2 x
Multiply the last terms. (1)( 5) 5
Add the first, outer, inner, and last terms,
and identify the like terms.
Combine like terms. 6 x 2 13 x 5
■ YOUR TURN
Multiply (2 x 3)(5 x 2).
(3 x + 1)(2 x - 5) = 6 x 2 - 15 x + 2 x - 5
■
Answer: 10 x 2 19 x 6
Study Tip
When the binomials are of the form
( ax b )( cx d ), the outer and inner
terms will be like terms and can be
combined.

EXAMPLE 7
Multiplying Binomials Resulting in Special Products
Find the following: a. ( x 5)( x 5) b. ( x 5) 2 c. ( x 5) 2
Solution:
a. ( x - 5)( x + 5) = x
2
+ 5 x - 5 x - 5
2
= x
2
- 5
2
= x
2
- 25
32
CHAPTER 0
Prerequisites and Review
Square of a binomial sum: ( a b )
2
( a b )( a b ) a
2
2 ab b
2
Square of a binomial difference: ( a b )
2
( a b )( a b ) a
2
2 ab b
2
P
ERFECT SQUARES
( a + b )( a - b ) = a 2 - b 2
D
IFFERENCE OF TWO SQUARES
EXAMPLE 8
Finding the Square of a Binomial Sum
Find ( x 3)
2
.
INCORRECT
ERROR:
Don’t forget the middle term, which is
twice the product of the two terms in
the binomial.
( x + 3) 2 Z x 2 + 9
CORRECT
( x 3)
2
( x 3)( x 3)
x
2
3 x 3 x 9
x 2 6 x 9
Forgetting the middle term, which is twice the product of the two terms in
the binomial.
C
O M M O N
M
I S TA K E
★
Let a and b be any real number, variable, or algebraic expression in the following
special products.
Some products of binomials occur frequently in algebra and are given special names.
Example 7 illustrates the difference of two squares and perfect squares .
r
First
r
Inner
s
Difference of two squares
r
Outer
r
Last
r
First
r
Inner
r
Outer
r
First
r
Inner
r
Outer
r
Last
r
Last
b.