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Blitzer, Intermediate Algebra, 5e – Slide #3 Section 5.2 Multiplying Polynomials Check Point 1a Multiply Check Point 1b Multiply
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§ 5.2
Multiplication of Polynomials
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 5.2
Multiplying PolynomialsEXAMPLE
SOLUTIONMultiply
yxyx 724 36
. 36 724 yxyx
127436 yyxx Rearrange factors
Multiply coefficients andadd exponents
127418 yx
31118 yx Simplify
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 5.2
Multiplying PolynomialsCheck Point 1a
Multiply . 36 4275 yxyx
Check Point 1bMultiply . 310 236634 zyxzyx
. 18 117 yx
. 30 8610 zyx
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 5.2
Multiplying PolynomialsEXAMPLE
SOLUTIONMultiply . 7536 253 xxx
7536 253 xxx
765636 32353 xxxxx Distribute
Multiply coefficients andadd exponents
358 423018 xxx
Check Point a and b p 318
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 5.2
Multiplying PolynomialsEXAMPLE
SOLUTIONMultiply . 22 bababa
22 bababa
2222 bababbabaa Multiply the trinomial by each term of the binomial
Multiply coefficients andadd exponents
2222 bbabbabbaabaaa 322223 babbaabbaa
33 ba
Distribute
Simplify
Note this: We multiply each term of the binomial by each term of the trinomial. We get 6 products in all.
Compare by DOING VERTICALLY, Do Check Point 3 or Check Point 4
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 5.2
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
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 5.2
Multiplying Polynomials - FOILEXAMPLE
SOLUTIONMultiply . 1534 xx
1534 xx 13531454 xxxx
Combine like terms
Multiply315420 2 xxx
31920 2 xx
F O I Lfirst
outside
inside
last
Do Check Point 5 on page 321
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 5.2
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
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 5.2
Multiplying Polynomials – Special Formulas
EXAMPLE
SOLUTIONMultiply .4 2yx
24 yx
222 2 BABABA
Use the special-product formula shown.
+ + = Product
+ +
2
TermFirst
Terms theof
Product22
TermLast
24x yx42 2y 22 816 yxyx
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 5.2
Multiplying Polynomials – Special Formulas
EXAMPLE
SOLUTIONMultiply . 43 2yx
243 yx
222 2 BABABA
Use the special-product formula shown.
- + = Product
- +
2
TermFirst
Terms theof
Product22
TermLast
23x yx 432 24y 22 16249 yxyx
Do Check Points 6 and 7 on pages 322-3
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 5.2
Multiplying Polynomials – Special Formulas
EXAMPLE
SOLUTIONMultiply . 4343 22 yxyyxy
222 4 3 yxy
22 BABABA
Use the special-product formula shown.
First Term Squared
Second Term Squared
Product- =
= 242 169 yyx
Do Check Point 8 on page 324
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 5.2
Multiplying Polynomials – Special Formulas
EXAMPLE
SOLUTIONMultiply . 33 yxyx
92 22 yyxx
2A
We can group the terms so that the formula for the square of a binomial can be applied.
2B(A + B) (A - B) = - 22 333 yxyxyx
92 22 yxyx
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 5.2
Multiplying Polynomials p 324
Check Point 9aMultiply yxyx 523523
Check Point 9bMultiply 232 yx
. 254129 22 yxx
961244 22 yxyxyx
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 5.2
Multiplying Polynomial FunctionsEXAMPLE
SOLUTION
following. theofeach Find .10 and 4Let xxgxxf
4062 xx
(a) (fg)(x) = f (x) g(x)
404102 xxx
(a) (fg)(x)(b) (fg)(-1)(c) (fg)(0)
= (x - 4)(x + 10)F O I L
. 4062 xxxfgThus,
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 5.2
Multiplying Polynomial Functions
45401611 2 fg
CONTINUED
(b) We use the product function to find (fg)(-1) – that is, the value of the function fg at -1. Replace x with -1.
40400600 2 fg
(c) We use the product function to find (fg)(0) – that is, the value of the function fg at 0. Replace x with 0.
Blitzer, Intermediate Algebra, 5e – Slide #16 Section 5.2
Multiplying Polynomial FunctionsEXAMPLE
SOLUTION
afhaf
xxxf
:following hesimplify t and find ,154Given 2
To find f (a + h) – f (a), we first replace each occurrence of x with a + h and then replace each occurrence of x with a. Then we perform the resulting operations and simplify.
154154 22 aahaha
f (a + h) – f (a)
Use , replacing x with a + h and a, respectively.
154 2 xxxf
Blitzer, Intermediate Algebra, 5e – Slide #17 Section 5.2
Multiplying Polynomial Functions
154155484 222 aahahaha
Combine like terms
15415524 222 aahahaha Multiply as indicated
CONTINUED
Multiply
hhah 548 2
Look at Check Point 11 on page 326
DONE
