### 6-2

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6-2 Multiplying Polynomials
Warm Up
Lesson Presentation
Lesson Quiz
Holt
McDougal
Algebra
Holt
McDougal
Algebra
22
6-2
Multiplying Polynomials
Warm Up
Multiply.
1. x(x3)
2. 3x2(x5)
3. 2(5x3)
4. x(6x2)
5. xy(7x2)
6. 3y2(–3y)
Holt McDougal Algebra 2
6-2
Multiplying Polynomials
Objectives
Multiply polynomials.
Use binomial expansion to expand
binomial expressions that are raised to
positive integer powers.
Holt McDougal Algebra 2
6-2
Multiplying Polynomials
To multiply a polynomial by a monomial, use
the Distributive Property and the Properties
of Exponents.
Holt McDougal Algebra 2
6-2
Multiplying Polynomials
Example 1: Multiplying a Monomial and a Polynomial
Find each product.
A. 4y2(y2 + 3)
B. fg(f4 + 2f3g – 3f2g2 + fg3)
Holt McDougal Algebra 2
6-2
Multiplying Polynomials
Check It Out! Example 1
Find each product.
a. 3cd2(4c2d – 6cd + 14cd2)
b. x2y(6y3 + y2 – 28y + 30)
Holt McDougal Algebra 2
6-2
Multiplying Polynomials
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.
Holt McDougal Algebra 2
6-2
Multiplying Polynomials
Example 2A: Multiplying Polynomials
Find the product.
(a – 3)(2 – 5a + a2)
Method 1 Multiply horizontally.
Holt McDougal Algebra 2
6-2
Multiplying Polynomials
Example 2A: Multiplying Polynomials
Find the product.
(a – 3)(2 – 5a + a2)
Method 2 Multiply vertically.
Holt McDougal Algebra 2
6-2
Multiplying Polynomials
Example 2B: Multiplying Polynomials
Find the product.
(y2 – 7y + 5)(y2 – y – 3)
Multiply each term of one polynomial by each term of
the other. Use a table to organize the products.
y2
–y
–3
The top left corner is the first
2
4
3
2
y
y
–y –3y
term in the product. Combine
terms along diagonals to get
–7y –7y3 7y2 21y
the middle terms. The bottom
right corner is the last term in
5
2
5y
–5y –15 the product.
y4 + (–7y3 – y3 ) + (5y2 + 7y2 – 3y2) + (–5y + 21y) – 15
y4 – 8y3 + 9y2 + 16y – 15
Holt McDougal Algebra 2
6-2
Multiplying Polynomials
Check It Out! Example 2a
Find the product.
(3b – 2c)(3b2 – bc – 2c2)
Holt McDougal Algebra 2
6-2
Multiplying Polynomials
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.
x2
–4x
1
The top left corner is the first
2
4
3
2
x
x
–4x
x
term in the product. Combine
terms along diagonals to get
2
5x 5x3 –20x
5x
the middle terms. The bottom
right corner is the last term in
–2 –2x2 8x
–2
the product.
x4 + (–4x3 + 5x3) + (–2x2 – 20x2 + x2) + (8x + 5x) – 2
x4 + x3 – 21x2 + 13x – 2
Holt McDougal Algebra 2
6-2
Multiplying Polynomials
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.
Holt McDougal Algebra 2
6-2
Multiplying Polynomials
Solve A(p) = l(p)  w(p).
Holt McDougal Algebra 2
Solve V(p) = A(p)  h(p).
6-2
Multiplying Polynomials
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)
Holt McDougal Algebra 2
6-2
Multiplying Polynomials
Check It Out! Example 3
Multiply the two polynomials.
Holt McDougal Algebra 2
6-2
Multiplying Polynomials
Example 4: Expanding a Power of a Binomial
Find the product.
(a + 2b)3
Holt McDougal Algebra 2
6-2
Multiplying Polynomials
Check It Out! Example 4a
Find the product.
(x + 4)4
Holt McDougal Algebra 2
6-2
Multiplying Polynomials
Check It Out! Example 4b
Find the product.
(2x – 1)3
Holt McDougal Algebra 2
6-2
Multiplying Polynomials
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 Pascal’s 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.
Holt McDougal Algebra 2
6-2
Multiplying Polynomials
This information is formalized by the Binomial
Theorem, which you will study further in Chapter 11.
Holt McDougal Algebra 2
6-2
Multiplying Polynomials
Example 5: Using Pascal’s Triangle to Expand
Binomial Expressions
Expand each expression.
A. (k – 5)3
B. (6m – 8)3
Holt McDougal Algebra 2
6-2
Multiplying Polynomials
Check It Out! Example 5
Expand each expression.
a. (x + 2)3
b. (x – 4)5
Holt McDougal Algebra 2
6-2
Multiplying Polynomials
Check It Out! Example 5
Expand the expression.
c. (3x + 1)4
Holt McDougal Algebra 2
6-2
Multiplying Polynomials
Lesson Quiz
Find each product.
1. 5jk(k – 2j)
2. (2a3 – a + 3)(a2 + 3a – 5)
3. The number of items is modeled by
0.3x2 + 0.1x + 2, and the cost per item is
modeled by g(x) = –0.1x2 – 0.3x + 5. Write a
polynomial c(x) that can be used to model the
total cost.
4. Find the product.
(y – 5)4
5. Expand the expression.
(3a – b)3
Holt McDougal Algebra 2
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