Multiplying Polynomials

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6-2
Multiplying Polynomials
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2
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Warm Up Multiply.
1. x(x3)
x4
2. 3x2(x5)
3x7
3. 2(5x3)
10x3
4. x(6x2)
6x3
5. xy(7x2)
7x3y
6. 3y2(3y)
9y3
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Objectives
Multiply polynomials. Use binomial expansion to
expand binomial expressions that are raised to
positive integer powers.
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To multiply a polynomial by a monomial, use the
Distributive Property and the Properties of
Exponents.
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Check It Out! Example 1
Find each product.
a. 3cd2(4c2d 6cd 14cd2)
3cd2(4c2d 6cd 14cd2)
Distribute.
3cd2 ? 4c2d 3cd2 ? 6cd 3cd2 ? 14cd2
12c3d3 18c2d3 42c2d4
Multiply.
b. x2y(6y3 y2 28y 30)
x2y(6y3 y2 28y 30)
Distribute.
x2y ? 6y3 x2y ? y2 x2y ? 28y x2y ? 30
6x2y4 x2y3 28x2y2 30x2y
Multiply.
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To multiply any two polynomials, use the
Distributive Property and multiply each term in
the second polynomial by each term in the first.
Keep in mind that if one polynomial has m terms
and the other has n terms, then the product has
mn terms before it is simplified.
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Check It Out! Example 2a
Find the product.
(3b 2c)(3b2 bc 2c2)
Multiply horizontally.
Write polynomials in standard form.
(3b 2c)(3b2 2c2 bc)
Distribute 3b and then 2c.
3b(3b2) 3b(2c2) 3b(bc) 2c(3b2) 2c(2c2)
2c(bc)
Multiply. Add exponents.
9b3 6bc2 3b2c 6b2c 4c3 2bc2
9b3 9b2c 4bc2 4c3
Combine like terms.
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Check It Out! Example 2b
Find the product.
(x2 4x 1)(x2 5x 2)
Multiply each term of one polynomial by each term
of the other. Use a table to organize the
products.
The top left corner is the first term in the
product. Combine terms along diagonals to get the
middle terms. The bottom right corner is the last
term in the product.
x4 4x3 x2
5x3 20x2 5x
2x2 8x 2
x4 (4x3 5x3) (2x2 20x2 x2) (8x
5x) 2
x4 x3 21x2 13x 2
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Check It Out! Example 3
Mr. Silva manages a manufacturing plant. From
1990 through 2005 the number of units produced
(in thousands) can be modeled by N(x) 0.02x2
0.2x 3. The average cost per unit (in dollars)
can be modeled by C(x) 0.004x2 0.1x 3.
Write a polynomial T(x) that can be used to model
the total costs.
Total cost is the product of the number of units
and the cost per unit.
T(x) N(x) ? C(x)
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Check It Out! Example 3
Multiply the two polynomials.
0.02x2 0.2x 3
? 0.004x2 0.1x 3
0.06x2 0.6x 9
0.002x3 0.02x2 0.3x
0.00008x4 0.0008x3 0.012x2
0.00008x4 0.0028x3 0.028x2 0.3x 9
Mr. Silvas total manufacturing costs, in
thousands of dollars, can be modeled by T(x)
0.00008x4 0.0028x3 0.028x2 0.3x 9
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Check It Out! Example 4a
Find the product.
(x 4)4
(x 4)(x 4)(x 4)(x 4)
Write in expanded form.
Multiply the last two binomial factors.
(x 4)(x 4)(x2 8x 16)
Multiply the first two binomial factors.
(x2 8x 16)(x2 8x 16)
Distribute x2 and then 8x and then 16.
x2(x2) x2(8x) x2(16) 8x(x2) 8x(8x)
8x(16) 16(x2) 16(8x) 16(16)
Multiply.
x4 8x3 16x2 8x3 64x2 128x 16x2 128x
256
Combine like terms.
x4 16x3 96x2 256x 256
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Check It Out! Example 4b
Find the product.
(2x 1)3
(2x 1)(2x 1)(2x 1)
Write in expanded form.
(2x 1)(4x2 4x 1)
Multiply the last two binomial factors.
Distribute 2x and then 1.
2x(4x2) 2x(4x) 2x(1) 1(4x2) 1(4x)
1(1)
8x3 8x2 2x 4x2 4x 1
Multiply.
8x3 12x2 6x 1
Combine like terms.
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Notice the coefficients of the variables in the
final product of (a b)3. these coefficients are
the numbers from the third row of Pascal's
triangle.
Each row of Pascals triangle gives the
coefficients of the corresponding binomial
expansion. The pattern in the table can be
extended to apply to the expansion of any
binomial of the form (a b)n, where n is a whole
number.
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This information is formalized by the Binomial
Theorem, which you will study further in Chapter
11.
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Check It Out! Example 5
Expand each expression.
a. (x 2)3
1 3 3 1
Identify the coefficients for n 3, or row 3.
1(x)3(2)0 3(x)2(2)1 3(x)1(2)2
1(x)0(2)3
x3 6x2 12x 8
b. (x 4)5
1 5 10 10 5 1
Identify the coefficients for n 5, or row 5.
1(x)5(4)0 5(x)4(4)1 10(x)3(4)2
10(x)2(4)3 5(x)1(4)4 1(x)0(4)5
x5 20x4 160x3 640x2 1280x 1024
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Check It Out! Example 5
Expand the expression.
c. (3x 1)4
1 4 6 4 1
Identify the coefficients for n 4, or row 4.
1(3x)4(1)0 4(3x)3(1)1 6(3x)2(1)2
4(3x)1(1)3 1(3x)0(1)4
81x4 108x3 54x2 12x 1