2-4 Zeros of a Polynomial Function

1. Chapter 2 Power, Polynomial, and Rational Functions 2-4Zeros of a Polynomial Function

2. Warm-up Factor using long division. 1. Find f(c) using synthetic substitution. 2.

3. Homework Check • …and a short Homework Quiz

4. Complete the following: • Recall the values of the following: i = i2 = i3 = i4 = i5 = i6 = i7 = • Fractions • Integers • Irrational numbers • Imaginary numbers • Natural numbers • Rational numbers • Real numbers • Whole numbers

5. Recall the values of the following: i = i2 = i3 = i4 = i5 = i6 = i7 = • Integers • Irrational numbers • Imaginary numbers • Natural numbers • Rational numbers • Real numbers • Whole numbers

6. Objectives for 2-4 • Find real zeros of polynomial functions • Find complex zeros of polynomial functions

7. 1. Real Zeros of a Polynomial • The leading coefficient and constant term with integer coefficients can be used to determine a list of all possible rational zeros. • Then you can determine actual zeros using synthetic division. • This is the Rational Zero Theorem

8. Rational Zero Theorem Every rational zero of a polynomial has the form , where • p is an integer factor of the constant term • q is an integer factor of the leading coefficient

9. Example 1: List all possible rational zeros. Then determine which, if any, are zeros.

10. Example 2: List all possible rational zeros. Then determine which, if any, are zeros.

11. Example 3: List all possible rational zeros. Then determine which, if any, are zeros.

12. 2. Writing a Polynomial Given its Zeros Write a polynomial function of least degree with real coefficients in standard form that has -1, 2, and 2 – i as zeros.

13. That’s enough for one day… • Practice these skills, and then we will put everything together after the mid-chapter quiz next class.

14. The Mid-chapter quiz on Tuesday… Topics covered (2 – 1 through 2 – 3): • Domain and Range of graphed functions • Solving radical equations • Determining end behavior of a polynomial without the use of a calculator • Determining the number of turning points and where functions increase and decrease • Using long division, synthetic division, and synthetic substitution to determine factors of polynomial functions

15. Assignment due Thursday • Practice with skills from today’s lesson p. 127, #3, 5, 11, 13, 15, 33, 35, 37.

16. Finish the Mid-chapter quiz

17. Data analysis

18. Almost 14% of the variability in overall grade can be attributed to the homework grade.

19. Almost half of the variability in overall grade can be attributed to your homework average.

20. Homework Check:

21. A couple more concepts will be of some help in pulling all the ideas together in this section. • Upper and Lower Bounds Tests (EASY!) • Descartes’ Rule of Signs (non-essential) • Pull it all together with the first half of this lesson

22. Using the Upper and Lower Bounds Test • To narrow the search for real zeros, you can determine the interval in which the real zeros are located. • This function seems have real zeros between -2 and 2. Eliminate all zeros outside that interval. • How easy is that!? Lower Bound Upper Bound

23. Descartes’ Rule of Signs tells you the number of positive or negative real zeros • If you are interested, you can find this on p. 123 in your textbook. • It is non-essential, but an interesting theoretical construct by the great French mathematician Decartes.

24. Pulling it all together! • Factor and find the zeros (both real and irreducible quadratic factors) • Then, factor the irreducible quadratic factors into imaginary roots and list all the zeros.

25. Example: • Write k(x) as the product of linear and irreducible quadratic factors. • Write k(x) as the product of linear factors. • List all the zeros of k(x).

26. Example:Write k(x) as the product of linear and irreducible quadratic factors. • Step 1: List all possible factors

27. Example:Write k(x) as the product of linear and irreducible quadratic factors. • Step 2 (optional) Check Descartes Rule of Signs

28. Example:Write k(x) as the product of linear and irreducible quadratic factors. • Step 3: Look at the graph in your calculator and find upper and lower bounds of the real roots. • Eliminate all possible roots outside of the upper and lower bounds. • Start testing with those.

29. Example:Write k(x) as the product of linear and irreducible quadratic factors. • The graph suggests that 4 is a zero. Start there. • Use the depressed polynomial to test the next possible zero.

30. Example:Write k(x) as the product of linear and irreducible quadratic factors. • The graph suggests that -2 is another zero. Try that one. • Use the depressed polynomial to test the next possible zero.

31. Example:Write k(x) as the product of linear and irreducible quadratic factors. • The graph suggests that -3 is another zero. Try that one. • Write the depressed polynomial. Note that it is irreducible (It can’t be factored with real roots).

32. Summarize all the roots and factors • Roots: 4, -2, and -3 • Factors:

33. Assignment: p. 127, 39 – 47 odds