Transcript Lesson 3.2B

Multiplying Polynomials
• How do we multiply polynomials?
•How do we use binomial expansion to
expand binomial expressions that are
raised to positive integer powers?
Holt McDougal Algebra 2
Multiplying Polynomials
Example 1: Business Application
A standard Burly Box is p ft by 3p ft by 4p ft. A large Burly
Box has 1.5 ft added to each dimension. Write a polynomial
V(p) in standard form that can be used to find the volume of a
large Burly Box.
The volume of a large Burly Box is the product of the area of the
base and height. V  p   A  p   h  p 
The area of the base of the large Burly Box is the product of the
length and width of the box.
A p   l p  w p 
The length, width, and height of the large Burly Box are greater
than that of the standard Burly Box.
l  p   p  1 .5
Holt McDougal Algebra 2
w  p   3 p  1 .5
h  p   4 p  1 .5
Multiplying Polynomials
Example 1: Business Application
Solve A(p) = l(p)  w(p).
p + 1.5
 3p + 1.5
1 .5 p  2 . 2 5
2
3 p  4 .5 p
2
3 p  6 p  2 .2 5
Solve V(p) = A(p)  h(p).
3p2 + 6p + 2.25
 4p + 1.5
4 .5 p  9 p  3 . 3 7 5
3
2
12 p  24 p  9 p
2
3
12 p  2 8 . 5 p  1 8 p  3 . 3 7 5
2
The volume of a large Burly Box can be modeled by
V(p) = 12p3 + 28.5p2 + 18p + 3.375
Holt McDougal Algebra 2
Multiplying Polynomials
Example 2: Business Application
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 
Multiply the two polynomials.
Holt McDougal Algebra 2
Multiplying Polynomials
Example 2: Business Application
0.02x2 + 0.2x + 3
 –0.004x2 – 0.1x + 3
0 .0 6 x  0 .6 x  9
3
2
 0 .0 0 2 x  0 .0 2 x  0 .3 x
4
 0 .0 0 0 0 8 x  0 .0 0 0 8 x 3  0 .0 1 2 x 2
 0 . 0 0 0 0 8 x 4  0 . 0 0 2 8 x 3 0 . 0 2 8 x 2  0 . 3 x  9
2
Mr. Silva’s total manufacturing costs, in thousands of
dollars, can be modeled by
T(x) = –0.00008x4 – 0.0028x3 + 0.028x2 + 0.3x + 9
Holt McDougal Algebra 2
Multiplying Polynomials
Example 3: Expanding a Power of a Binomial
Find the product.
a  2 ab  2 ab  4b
(a +
Write in expanded form.
(a + 2b)(a + 2b)(a + 2b)
Multiply the last two
2
2
(a + 2b)(a + 4ab + 4b )
binomial factors.
2b)3
2
2
Distribute a and then 2b.
a  4 a b  4 a b  2 a b  8 a b  8b
3
Combine like terms.
a  6 a 2 b  1 2 a b 2  8b 3
3
2
Holt McDougal Algebra 2
2
2
2
3
Multiplying Polynomials
Example 4: Expanding a Power of a Binomial
Find the product.
x  4 x  4 x  16
(x +
Write in expanded form.
(x + 4)(x + 4)(x + 4)(x + 4)
Multiply the last two
2
(x + 4)(x + 4)(x + 8x + 16)
binomial factors.
2
4)4
Multiply the first two
binomial factors.
Distribute x2 and then 8x and then 16.
(x2 + 8x + 16)(x2 + 8x + 16)
x  8x  16 x  8x  64 x  128 x  16 x  128 x  256
4
3
2
3
2
x  16 x  96 x  2 5 6 x  256
4
3
2
Holt McDougal Algebra 2
2
Combine like terms.
Multiplying Polynomials
Example 5: Expanding a Power of a Binomial
Find the product.
(2x – 1)3
(2x – 1)(2x – 1)(2x – 1)
(2x – 1)(4x2 – 4x + 1)
Write in expanded form.
Multiply the last two
binomial factors.
Distribute 2x and then –1.
8x  8 x  2 x  4 x  4 x  1
3
2
2
8x  12 x  6 x  1
3
Holt McDougal Algebra 2
2
Combine like terms.
Multiplying Polynomials
Lesson 3.2 Practice B
Holt McDougal Algebra 2