1. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.2, Slide 1 Chapter 6 Polynomial Functions

2. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.2, Slide 2 6.2 Multiplying Polynomial Expressions and Functions

3. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.2, Slide 3 Monomials, Binomials, and Trinomials A polynomial is a monomial, binomial, or trinomial, depending on the number of terms.

4. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.2, Slide 4 Example: Finding the Product of Two Monomials Find the product

5. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.2, Slide 5 Solution

6. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.2, Slide 6 Equivalent Expressions Warning The left expression is a product, whereas the right expression is a difference.

7. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.2, Slide 7 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.

8. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.2, Slide 8 Example: Finding the Products of Two Polynomials Find the product. 1. (2x + y)(5x 2 – 3xy + 4y 2 ) 2. (x 2 – 3x + 2)(x 2 + x – 5)

9. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.2, Slide 9 Solution 1. Multiply each term in the first polynomial by each term in the second polynomial:

10. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.2, Slide 10 Solution 2.

11. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.2, Slide 11 Solution 2. Use a graphing calculator table to verify our work.

12. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.2, Slide 12 Squaring a Binomial (A + B) 2 = A 2 + 2AB + B 2 (A – B) 2 = A 2 – 2AB + B 2 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.

13. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.2, Slide 13 Example: Simplifying Squares of Binomials Simplify. 1. (x + 7) 2 2. (5t – 4w) 2 3. (3r 2 + 2y 2 ) 2

14. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.2, Slide 14 Solution 1. Substitute x for A and 7 for B: Or,

15. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.2, Slide 15 Solution 2. Substitute 5t for A and 4w for B: Or,

16. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.2, Slide 16 Solution 3.

17. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.2, Slide 17 Perfect-Square Trinomial A perfect-square trinomial is a trinomial equivalent to the square of a binomial.

18. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.2, Slide 18 Product of Binomial Conjugates (A + B)(A – B) = A 2 – B 2 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.

19. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.2, Slide 19 Example: Multiplying Binomial Conjugates Find the product. 1. (x + 6)(x + 6) 2. (3p – 8q)(3p + 8q) 3. (4m 2 – 7rt)(4m 2 + 7rt) 4. (x + 3)(x – 3)(x 2 + 9)

20. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.2, Slide 20 Solution 1. Substitute x for A and 6 for B: 2. Substitute 3p for A and 8q for B:

21. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.2, Slide 21 Solution 3. 4.

22. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.2, Slide 22 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)

23. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.2, Slide 23 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.

24. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.2, Slide 24 Example: Using a Product Function to Model a Situation

25. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.2, Slide 25 Example: Using a Product Function to Model a Situation 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?

26. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.2, Slide 26 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.

27. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.2, Slide 27 Solution 2. 3. The units of the expressions are millions of dollars.

28. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.2, Slide 28 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.

29. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.2, Slide 29 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.