1. 2.5 Zeros of Polynomial Functions Question to be answered: How do we determine zeros of functions that we cannot factor easily? Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 1

2. Review what you know… How many roots does have? Show that 1 is a root of this function. Find the other roots of this function. Write the function as the product of its factors. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2

3. Try another one. How many roots does have? Show that -1 is a root of this function. Find the other roots of this function. Write the function as the product of its factors. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3

4. Here’s the problem… How many roots does have? Show that 1 is a root of this function. Find the other roots of this function. When the polynomial is more than degree 2, we need some help finding the roots. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4

5. Rational Zero Test This test allows you to find all POSSIBLE rational roots Rational – numbers that can be written as fractions You then need to determine which ones are actually roots. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5

6. Rational Zero Test Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 Rational Zero Test: If a polynomial f(x) has integer coefficients, every rational zero of f has the form where p and q have no common factors other than 1. Example: Find the rational zeros of f(x) = x 3 + 3x 2 – x – 3. The possible rational zeros are ±1 and ±3. Synthetic division shows that the factors of f are (x + 3), (x + 1), and (x – 1). p is a factor of the constant term. q is a factor of the leading coefficient. q = 1 p = – 3 The zeros of f are – 3, – 1, and 1.

7. Using the Rational Zero Test Step 1: Find the factors of p. Step 2: Find the factors of q. Step 3: Find all combinations of p/q. Step 4: Use synthetic division to determine which are roots. Step 5: Factor the quadratic. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 Example: Find the rational zeros of f(x) = x 3 + 3x 2 – x – 3. q = 1 p = – 3

8. Why do we use this? Without this, we would have no where to start looking for zeros. We would be synthetically dividing everything. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8

9. Try it. Determine all the zeros of Use the Rational Zero Test Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9

10. A little different Determine all of the zeros of Use the Rational Zero Test Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10

11. Try Some… Page 179 #1-6 and 11-19 ODD Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11

12. Before Determine all of the possible rational roots and then all of the actual roots of Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12

13. Tips There are some tricks that we can use that would help us to eliminate some of the possible roots if our list is really long. Descartes’s Rule of Signs Lower and Upper Bounds Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13

14. Descartes’s Rule of Signs Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14

15. Upper and Lower Bounds Let f(x) be a polynomial with real coefficients and a positive leading coefficient. Suppose f(x) is divided by x-c, using synthetic division. If c>0 and each number in the last row is either positive or zero, c is an upper bound for the real zeros of f If c<0 and the numbers in the last row are alternately positive and negative (zero entries count as positive or negative), c is a lower bound for the real zeros of f Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15

16. Example: Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16

17. Use everything you have learned… Graph this function. Find all zeros of this function. Use the leading coefficient test to determine end behavior. f(x) = 2x 4 – 17x 3 + 35x 2 + 9x – 45. Check your work. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 17

18. Homework After: Why do we use the rational zero test? Homework: Page 180 #40-42, 55, 59, 60, 69 Test on Tuesday! Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 18