2. Polynomial Functions End Behavior Section 2-2

3. 2 Objectives I can determine if an equation is a polynomial in one variable I can find the degree of a polynomial I can use the Leading Coefficient Test for end behavior in Limit Notation

4. 3 A polynomial function is a function of the form where n is a nonnegative integer and each a i (i = 0, , n) is a real number. The polynomial function has a leading coefficient a n and degree n. Examples: Find the leading coefficient and degree of each polynomial function. Polynomial FunctionLeading Coefficient Degree – 2 5 1 3 14 0

5. 4 Complex Numbers Real NumbersImaginary Numbers RationalsIrrational

6. 5 Polynomial Functions/Equations: A polynomial function in one variable may look like this. A.The coefficients are complex numbers (real or imaginary). B. Exponents must be a non-negative integer (zero or positive). C.The leading coefficient (the coefficient of the variable with greatest degree) may not be zero.

7. 6 Polynomials Not a polynomial the exp. is not an integer the exp. is not non-negative denominator has a variable factor

8. 7 EX:Determine if each expression is a polynomial in one variable YES No

9. 8 Practice 1: Given the following equations determine the following: 1.Determine if the equation a polynomial. Why or why not? 2. If the equation is a polynomial what is the degree of each term, of the polynomial. A. B. C. D. Yes, notice powers on the x are positive integers and coefficients are real numbers. No, notice power on the x is the fraction 1/2 No, notice power on x is -1 Yes, notice powers on the x are positive integers and coefficients are real numbers.

10. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 Group Exploration Directions: Divide into groups of 2 Open your text to page 141. Read the Exploration exercise 10 minutes!! Answers on next slides

11. 10 Group Exploration Open the text to page 141. Read the Exploration exercise instructions. Use the Leading coefficient test on page 141 a. b. c. d. e. f. g. The leading coefficient is + 1 The degree of the function is 3, that is, f(x) is a cubic.

12. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 Group Exploration Open the text to page 141. Read the Exploration exercise instructions. Use the Leading coefficient test on page 141 b. c. d. e. f. g. The leading coefficient is + 2 The degree is 5 and odd.

13. 12 Group Exploration Open the text to page 141. Read the Exploration exercise instructions. Use the Leading coefficient test on page 141 c. d. e. f. g. The leading coefficient is - 2 The degree is 5 and odd.

14. 13 Group Exploration Open the text to page 141. Read the Exploration exercise instructions. Use the Leading coefficient test on page 141 d. e. f. g. The leading coefficient is - 1 The degree is 3 and odd.

15. 14 Group Exploration Open the text to page 141. Read the Exploration exercise instructions. Use the Leading coefficient test on page 141 e. f. g. The leading coefficient is + 2 The degree is 2 and even. f(x)  +  as x  -  f(x)  +  as x  + 

16. 15 Group Exploration Open the text to page 141. Read the Exploration exercise instructions. Use the Leading coefficient test on page 141 f. g. The leading coefficient is + 1 The degree is 4 and even. f(x)  +  as x  -  f(x)  +  as x  + 

17. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16 Group Exploration Open the text to page 141. Read the Exploration exercise instructions. Use the Leading coefficient test on page 141 g. The leading coefficient is + 1 The degree is 2 and even. f(x)  +  as x  -  f(x)  +  as x  + 

18. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 17 Leading Coefficient Test As x grows positively or negatively without bound, the value f (x) of the polynomial function f (x) = a n x n + a n – 1 x n – 1 + … + a 1 x + a 0 (a n  0) grows positively or negatively without bound depending upon the sign of the leading coefficient a n and whether the degree n is odd or even. x y x y n odd n even a n positive a n negative

19. 18 Example: Right-Hand and Left-Hand Behavior Example: Describe the right-hand and left-hand behavior for the graph of f(x) = –2x 3 + 5x 2 – x + 1. As, and as, Negative-2Leading Coefficient Odd3Degree x y f (x) = –2x 3 + 5x 2 – x + 1

20. 19 Closure:

21. 20 Zeros of a Function A real number a is a zero of a function y = f (x) if and only if f (a) = 0. Real Zeros of Polynomial Functions If y = f (x) is a polynomial function and a is a real number then the following statements are equivalent. 1. (a, 0) is a zero of f. 2. x = a is a solution of the polynomial equation f (x) = 0. 3. (x – a) is a factor of the polynomial f (x). 4. (a, 0) is an x-intercept of the graph of y = f (x).

22. 21 Solution or RootZero or X-interceptFactor

23. 22 Homework WS 3-3