copyright © 2015, 2008, 2011 pearson education, inc. section 6.2, slide 1 chapter 6 polynomial...

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.2, Slide 1

Chapter 6Polynomial Functions


6.2 Multiplying Polynomial Expressions and Functions


Monomials, Binomials, and Trinomials

A polynomial is a monomial, binomial, or trinomial, depending on the number of terms.


Example: Finding the Product of Two Monomials

Find the product 3 2 64 2 .x y xy


Solution

6 63 2 234 2 4( 2)x x xy y yx y 4 88 yx


Equivalent Expressions

Warning

3 2 6 3 2 64 2 4 2x y xy x y xy

The left expression is a product, whereas the right expression is a difference.


Multiplying Two Polynomials

To multiply two polynomials, multiply each term in the first polynomial by each term in the second polynomial. Then combine like terms if possible.


Example: Finding the Products of Two Polynomials

Find the product.

1. (2x + y)(5x2 – 3xy + 4y2)

2. (x2 – 3x + 2)(x2 + x – 5)


Solution

1. Multiply each term in the first polynomial by each term in the second polynomial:


Solution

2.2 2( )( 5)23 x xx x

2 2 22 2 3 535 3x x x x xx xx xx 22 2 2 5x x

4 3 2 3 2 25 3 3 15 2 2 10x x x x x x x x 2 24 23 3 15 5 2 13 3 02x x xx x x xx

234 2 17 016xx xx


Solution

2. Use a graphing calculator table to verify our work.


Squaring a Binomial

(A + B)2 = A2 + 2AB + B2

(A – B)2 = A2 – 2AB + B2

In words, the square of a binomial equals the first term squared, plus (or minus) twice the product of the two terms, plus the second term squared.


Example: Simplifying Squares of Binomials

Simplify.

1. (x + 7)2 2. (5t – 4w)2 3. (3r2 + 2y2)2


Solution

1. Substitute x for A and 7 for B:

Or, 2( 7) ( 7)( 7)x x x 2 7 97 4x x x 2 14 49x x


Solution

2. Substitute 5t for A and 4w for B:

Or, 2(5 4 ) (5 4 )(5 4 )t w t w t w 2 220 2025 16twt tw w 2 225 40 16t tw w


Solution

3.

2 22 22222 23 232 22 3y r r yr y

4 2 2 49 12 4r r y y


Perfect-Square Trinomial

A perfect-square trinomial is a trinomial equivalent to the square of a binomial.


Product of Binomial Conjugates

(A + B)(A – B) = A2 – B2

In words, the product of two binomial conjugates is the difference of the square of the first term and the square of the second term.


Example: Multiplying Binomial Conjugates

Find the product.

1. (x + 6)(x + 6)

2. (3p – 8q)(3p + 8q)

3. (4m2 – 7rt)(4m2 + 7rt)

4. (x + 3)(x – 3)(x2 + 9)


Solution

1. Substitute x for A and 6 for B:

2. Substitute 3p for A and 8q for B:


Solution

3.

4.

2 2 22 2( )( ) (7 74 4 4 7 )rt rm m t rtm

4 2 216 49m r t

2 2 2( 3)( 3)( 9) ( 9) )9 (xx x xx

2 22 9x 4 81x


Product Function

Definition

If f and g are functions and x is in the domain of both functions, then we can form the product function f ∙ g:

(f ∙ g)(x) = f(x) ∙ g(x)


Example: Using a Product Function to Model a Situation

The annual cost of state corrections (prisons and related costs) per person in the United States can be modeled by the function C(t) = 3.3t + 89, where C(t) is the annual cost (in dollars per person at t years since 1990. The U.S. population can be modeled by the function P(t) = 2.8t + 254, where P(t) is the population (in millions) at t years since 1990. Data is shown in the table on the next slide.



1. Check that the models fit the data well.2. Find an equation of the product function C ∙ P.3. Perform a unit analysis of the expression C(t) ∙ P(t).4. Find (C ∙ P)(28). What does it mean in this

situation?5. Use a graphing calculator graph to determine

whether the function C ∙ P is increasing, decreasing, or neither for values of t between 0 and 30. What does your result mean in this situation?


Solution

1. Check the fit of the cost model on the left and the fit of the population model on the right. The models appear to fit the data fairly well.


Solution

2.

3.

The units of the expressions are millions of dollars.

( ( )))( () CC t tP P t (3.3 89) (2.8 254)t t

2 838.2 249.24 9.2 22,606tt t 2 1087.49.24 22,606tt


Solution

4.

This means the total cost of state corrections will be about $60,297 million ($60.297 billion) in 2018, according to the model.

2( )( ) 9.24 1087.4 22,6028 28 28 6C P

60,297


Solution

5. To graph the model, set the window as shown. For values of t between 0 and 30, the model is increasing. This means the total cost of state corrections has been increasing since 1990 and will continue to increase until 2020.

copyright © 2015, 2008, 2011 pearson education, inc. section 6.2, slide 1 chapter 6 polynomial...

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