1. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.1, Slide 1 Chapter 6 Polynomial Functions

2. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.1, Slide 2 6.1 Adding and Subtracting Polynomial Expressions and Functions

3. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.1, Slide 3 Polynomials A term is a constant, a variable, or a product of a constant and one or more variables raised to powers. Examples: A monomial is a constant, a variable, or a product of a constant and one or more variables raised to counting-number powers. Examples: Definitions

4. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.1, Slide 4 Polynomials Definitions A polynomial, or polynomial expression, is a monomial or a sum of monomials. Examples: 5x 3 – 2x 2 + 7x – 4, 4x 5 y 2 – x 2, 4x + 1, 5, x, –2x 3 We write polynomials in descending order. The degree of a term in one variable is the exponent on the variable. 7x 4 is degree 4.

5. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.1, Slide 5 Polynomials Definition The degree of a polynomial is the largest degree of any nonzero term of the polynomial. 4x 5 – 9x 3 + 1 has degree 5.

6. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.1, Slide 6 Example: Describing Polynomials Use words such as linear, quadratic, cubic, polynomial, degree, one variable, and two variables to describe the expression. 1. –3x 2 + 8x – 4 2. 5x 3 – 2x 2 + 9x + 1 3. 6a 5 b 2 – 9a 3 b 3 – 2ab 4

7. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.1, Slide 7 Solution 1. –3x 2 + 8x – 4 The term –3x 2 has degree 2, which is larger than the degrees of the other terms. So, the expression is a quadratic (second-degree) polynomial in one variable. 2. 5x 3 – 2x 2 + 9x + 1 The terms 5x 3 has degree 3, which is larger than the degrees of the other terms. So, the expression is a cubic (third-degree) polynomial in one variable.

8. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.1, Slide 8 Solution 3. 6a 5 b 2 – 9a 3 b 3 – 2ab 4 The term 6a 5 b 2 has degree 7 (the sum of the exponents of the variables), which is larger than the degrees of the other terms. So, the expression is a seventh-degree polynomial in two variables.

9. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.1, Slide 9 Combining Like Terms Definitions The coefficient of a term is the constant factor of the term. Example: For the term –7x 3, the coefficient is –7. The leading coefficient of a polynomial is the coefficient of the term with the largest degree. Example: For 3x 3 + 6x 2 – 4x – 9, the leading coefficient is 3. The terms 6x 5 and 8x 2 are unlike terms (not like terms), because the exponents of x are different.

10. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.1, Slide 10 Combining Like Terms To combine like terms, add the coefficients of the terms.

11. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.1, Slide 11 Example: Combining Like Terms Combine like terms when possible. 1. 5x 3 – 4x 2 + 2x 3 – x 2 2. 3p 3 t 2 + p 2 t – 8p 3 t 2 + 7p 2 t

12. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.1, Slide 12 Solution 1. Rearrange the terms so the terms with x 3 are adjacent and the terms with x 2 are adjacent: 2.

13. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.1, Slide 13 Adding Polynomials To add polynomials, combine like terms.

14. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.1, Slide 14 Example: Adding Polynomials Find the sum

15. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.1, Slide 15 Solution

16. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.1, Slide 16 Example: Subtracting Polynomials Find the difference

17. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.1, Slide 17 Solution

18. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.1, Slide 18 Solution Use a graphing calculator table to verify the work.

19. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.1, Slide 19 Subtracting Polynomials To subtract polynomials, first distribute –1; then combine like terms.

20. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.1, Slide 20 Quadratic Function Definition A quadratic function is a function whose equation can be put into the form f(x) = ax 2 + bx + c where a ≠ 0. This form is called the standard form.

21. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.1, Slide 21 Quadratic Function The graph of a quadratic function is called a parabola. The minimum point or maximum point is called the vertex. The vertical line that passes through a parabola’s vertex is called the axis of symmetry. The part of the parabola that lies to the left of the axis of symmetry is the mirror reflection of the part that lies to the right. Parabolas are illustrated on the next slide.

22. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.1, Slide 22 Parabolas

23. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.1, Slide 23 Example: Using a Graph to Find Values of a Function A graph of a quadratic function f is sketched below. 1. Find f(4). 2. Find x when f(x) = –3. 3. Find x when f(x) = 5. 4. Find x when f(x) = 6.

24. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.1, Slide 24 Solution 1. The blue arrows show that the input x = 4 leads to the output y = 3. So, f(4) = 3.

25. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.1, Slide 25 Solution 2. The red arrows show that the output y = –3 originates from the two inputs x = –2 and x = 6. So, the values of x are –2 and 6.

26. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.1, Slide 26 Solution 3. The green arrows show that the output y = 5 originates from the single input x = 2. So, the value of x is 2. There is a single input because the vertex (2, 5) is the only point on the parabola that has a y-coordinate equal to 5.

27. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.1, Slide 27 Solution 4. No point of the downward-opening parabola is above the vertex, which has a y-coordinate of 5. So, there is no point on the parabola with f(x) = 6.

28. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.1, Slide 28 Cubic Function Definition A cubic function is a function whose equation can be put into the form f(x) = ax 3 + bx 2 + cx + d where a ≠ 0.

29. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.1, Slide 29 Example: Graphing a Cubic Function Sketch the graph of f(x) = x 3.

30. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.1, Slide 30 Solution List some input-output pairs of the function, then plot the corresponding points and sketch a curve.

31. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.1, Slide 31 Sum Function, Difference Function Definition If f and g are functions and x is in the domain of both functions, then we can form the following functions: Sum function f + g, where (f + g)(x) = f(x) + g(x) Difference function f – g, where (f – g)(x) = f(x) – g(x)

32. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.1, Slide 32 Modeling with Sum Functions and Difference Functions Suppose A and B represent quantities. Then A + B represents the sum of the quantities The difference A – B tells us how much more there is of one quantity than the other quantity.

33. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.1, Slide 33 The Meaning of the Sign of a Difference If a difference A – B is positive, then A is more than B. If a difference A – B is negative, then A is less than B.

34. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.1, Slide 34 Example: Using a Sum Function and a Difference Function to Model a Situation Women’s and men’s enrollments at U.S. colleges and universities are shown in the table on the next slide for various years. The enrollments (in millions) W(t) and M(t) for women and men, respectively, are modeled by the system below, where t is the number of years since 1990. W(t) = 0.014t 2 – 0.08t + 7.96 M(t) = 0.012t 2 – 0.12t + 6.65

35. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.1, Slide 35 Example: Using a Sum Function and a Difference Function to Model a Situation

36. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.1, Slide 36 Example: Using a Sum Function and a Difference Function to Model a Situation 1. Find an equation of the sum function W + M. 2. Perform a unit analysis of the expression W(t) + M(t). 3. Find (W + M)(27). What does it mean in this situation? 4. Find an equation of the difference function W – M. 5. Find (W – M)(27). What does it mean in this situation?

37. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.1, Slide 37 Solution 1. 2. For the expressions W(t) + M(t), we have The units of the expression are millions of students.

38. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.1, Slide 38 Solution 3. (W + M)(27) = 0.026(27) 2 – 0.20(27) + 14.61 ≈ 28.16 This means the total enrollment for women and men in 1990 + 27 = 2017 will be about 28.2 millions students, according to the model.

39. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.1, Slide 39 Solution 4. 5. (W – M)(27) = 0.002(27) 2 + 0.04(27) + 1.31 ≈ 3.85 This means in 2017 women’s enrollment will exceed men’s enrollment by about 3.9 million students, according to the model.