Transcript Elementary and Intermediate Algebra 6e

Chapter 4
Polynomials
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4.5 Multiplication of Polynomials
• Multiplying Monomials
• Multiplying a Monomial and a Polynomial
• Multiplying Any Two Polynomials
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To Multiply Monomials
To find an equivalent expression for the
product of two monomials, multiply the
coefficients and then multiply the
variables using the product rule for
exponents.
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Example
Multiply: a) (6x)(7x)
b) (5a)(a)
c) (8x6)(3x4)
Solution
a) (6x)(7x) = (6  7) (x  x)
= 42x2
b) (5a)(a) = (5a)(1a)
= (5)(1)(a  a)
= 5a2
c) (8x6)(3x4) = (8  3) (x6  x4)
= 24x6 + 4
= 24x10
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Example
Multiply: a) x and x + 7
b) 6x(x2  4x + 5)
Solution
a) x(x + 7) = x  x + x  7
= x2 + 7x
b) 6x(x2  4x + 5) = (6x)(x2)  (6x)(4x) + (6x)(5)
= 6x3  24x2 + 30x
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The Product of a Monomial and a
Polynomial
To multiply a monomial and a
polynomial, multiply each term of the
polynomial by the monomial.
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Example
Multiply: 5x2(x3  4x2 + 3x  5)
Solution
5x2(x3  4x2 + 3x  5)
= 5x5  20x4 + 15x3  25x2
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Example
Multiply each of the following.
a) x + 3 and x + 5
b) 3x  2 and x  1
Solution
a) (x + 3)(x + 5) = (x + 3)x + (x + 3)5
= x(x + 3) + 5(x + 3)
=xx+x3+5x+53
= x2 + 3x + 5x + 15
= x2 + 8x + 15
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Example
Solution
b) (3x  2)(x  1) = (3x  2)x  (3x  2)1
= x(3x  2)  1(3x  2)
= x  3x  x  2  1  3x  1(2)
= 3x2  2x  3x + 2
= 3x2  5x + 2
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The Product of Two Polynomials
To multiply two polynomials P and Q, select one of
the polynomials, say P. Then multiply each term of
P by every term of Q and combine like terms.
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Example
Multiply: (5x3 + x2 + 4x)(x2 + 3x)
Solution
5x3 + x2 + 4x
x2 + 3x
15x4 + 3x3 + 12x2
5x5 + x4 + 4x3
5x5 + 16x4 + 7x3 + 12x2
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Example
Multiply: (3x2  4)(2x2  3x + 1)
Solution
2x2  3x + 1
3x2
4
 8x2 + 12x  4
6x4 + 9x3  3x2
6x4 + 9x3  11x2 + 12x  4
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