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Chapter 5
Section 5
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
5.5
Multiplying Polynomials
1
Multiply a monomial and a polynomial.
2
Multiply two polynomials.
3
Multiply binomials by the FOIL method.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 1
Multiply a monomial and a
polynomial.
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Multiply a monomial and a polynomial.
As shown in Section 5.1, we find the product of two
monomials by using the rules for exponents and the
commutative and associative properties. For example
8m6  9n6   8  9   m6  n6   72m6 n6 .
To find the product of a monomial and a polynomial with
more than one term we use the distributive property and
multiplication of monomials.
Do not confuse addition of terms with multiplication of terms.
For instance,
7q5  2q5  9q5 , but
 7q  2q   7  2q
5
5
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
55
 14q10 .
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EXAMPLE 1
Multiplying Monomials and
Polynomials
Find the product.
2 x 4  3x 2  2 x  5 
Solution:
  2 x  3x    2 x  2 x    2 x  5 
4
2
 6 x  4 x  10 x
6
5
4
4
4
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Objective 2
Multiply two polynomials.
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Multiply two polynomials.
We can use the distributive property repeatedly to find the
product of any two polynomials. For example, to find the
product of the polynomials x2 + 3x +5 and x − 4, think of x − 4 as
a single quantity and use the distributive property as follows.
 x 2  3x  5  x  4   x 2  x  4   3x  x  4   5  x  4 
Now use the distributive property three more times to find
x2(x − 4), 3x(x − 4), and 5(x − 4).
 x2  x   x2  4  3x  x   3x  4  5  x   5  4
 x3   4 x 2   3x 2   12 x   5 x   20 
 x3  x 2  7 x  20
This example suggests the following rule.
To multiply polynomials, multiply each term of the second
polynomial by each term of the first polynomial and add the
products.
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EXAMPLE 2
Multiplying Two Polynomials
Multiply (m3 − 2m + 1)·(2m2 + 4m + 3).
Solution:
 m3  2m 2   m3  4m   m3  3   2m   2m 2    2m  4m 
  2m  3  1 2m 2   1 4m   1 3
 2m5  4m4  3m3   4m3    8m2    6m   2m 2  4m  3
 2m5  4m 4  m3  6m 2  2m  3
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EXAMPLE 3
Multiplying Polynomials Vertically
Multiply.
3x 2  4 x  5
x4
Solution:
12 x 2  16 x  20
3x3  4 x 2  5 x
3x3  16 x 2  11x  20
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EXAMPLE 4
Multiplying Polynomials with
Fractional Coefficients Vertically
Multiply.
5 x3  10 x 2  20
1 2 2
x 
5
5
3
2
Solution:
2x  4x  8
x5  2 x 4  0 x3  4 x 2
x5  2 x 4  2 x3  8
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EXAMPLE 4A
Multiplying Polynomials with a
Rectangle Model
Use the rectangle method to find the product
 4x  3 x  2 .
Solution:
4x 2
3x
8x
6
4 x 2  11x  6
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Objective 3
Multiply binomials by the FOIL
method.
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Multiply binomials by the FOIL method.
In algebra, many times the polynomials to be multiplied are
binomials. For these products, the FOIL method reduces the
rectangle method to a systematic approach without the rectangle.
A summary of the steps in the FOIL method follows.
Step 1: Multiply the two First terms of the binomials to get the
first term of the answer.
Step 2: Find the Outer product and Inner product and add them
(when possible) to get the middle term of the answer.
Step 3: Multiply the two Last terms of the binomials to get the
last term of the answer.
L  15
F  x2
 x  3 x  5
O  5x
I  3x
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EXAMPLE 5
Using the FOIL Method
Find the product by the FOIL method.
F x
L  12
2
 x  2 x  6
O  6x I  2x
Solution:
 x2  6 x  2 x  8
 x  8 x  12
2
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EXAMPLE 6
Using the FOIL Method
Multiply  5x  6 2 y  3 .
F  10xy
L  18
5x  6 2 y  3
O  15x
I  12 y
Solution:
 10 xy  15 x  12 y  18
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EXAMPLE 7
Using the FOIL Method
Find each product.
 4 y  x  2 y  3x 
Solution:
 8 y  12 xy  2 xy  3x
 8 y 2  14 xy  3x2
3
3x  x  2 2 x  1
2
2
 3x3  2 x 2  1x  4 x  2 
 3x3  2 x 2  3x  2 
 6x  9x  6x
5
4
3
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