1. Warm-Up: January 9, 2012

2. Homework Questions?

3. Zeros of Polynomial Functions
Section 2.5

4. Rational Zero Theorem
If f(x) is a polynomial with integer coefficients, then every possible rational zero is given by:
Factors of the constant term
Possible rational zeros =
Factors of the leading coefficient

5. Example 1 (like HW #1-8)
List all possible rational zeros for

6. You-Try #1 (like HW #1-8)
List all possible rational zeros for

7. Finding Zeros of Polynomial Functions
Use the rational zero theorem to find all the possible rational zeros.
Use a guess-and-check method and synthetic division to find a zero
Use the successful synthetic division to get a lower- degree polynomial that can then be solved to find the remainder of the zeros.

8. Example 3 (like HW #9-22)
Find all rational zeros of

9. You-Try #3 (like HW #9-22)
Find all rational zeros of

10. Warm-Up: January 10, 2012
Find all rational zeros of

11. Homework Questions?

12. Example 4 (like HW #9-22)
Solve

13. You-Try #4 (like HW #9-22)
Solve

14. Linear Factorization Theorem
An nth-degree polynomial has n complex roots
c1, c2, …, cn
Each root can be written as a factor, (x-ci)
An nth-degree polynomial can be expressed as the product of a nonzero constant and n linear factors:

15. Example 5 (like HW #23-28) Factor the polynomial
as the product of factors that are irreducible over the rational numbers
as the product of factors that are irreducible over the real numbers
in completely factored form, including complex (imaginary) numbers

16. You-Try #5 (like HW #23-28) Factor the polynomial
as the product of factors that are irreducible over the rational numbers
as the product of factors that are irreducible over the real numbers
in completely factored form, including complex (imaginary) numbers

17. Warm-Up: January 11, 2012 Find all zeros of
Hint: Start by factoring as we did in Example 5.

18. Homework Questions?

19. Finding a Polynomial When Given Zeros
Write the basic form of a factored polynomial:
Fill in each “c” with a zero
Fractions can be written with the denominator in front of the “x”
If a complex (imaginary) number is a zero, so is its complex conjugate
Multiply the factors together
Use the given point to find the value of an

20. Example 6 (like HW #29-36)
Find an nth degree polynomial function with real coefficients with the following conditions:
n = 4
Zeros = {-2, -1/2, i}
f(1) = 18

21. You-Try #6 (like HW #29-36)
Find an nth degree polynomial function with real coefficients with the following conditions:
n = 3
Zeros = 4, 2i
f(-1) = -50

22. Warm-Up: January 12, 2012
Simplify and write in standard form (refer to 2.1 notes if needed)

23. Homework Questions?

24. Descarte’s Rule of Signs
Let f(x) be a polynomial with real coefficients
The number of positive real zeros of f is either equal to the number of sign changes of f(x) or is less than that number by an even integer. If there is only one variation in sign, there is exactly one positive real zero.
The number of negative real zeros of f is either equal to the number of sign changes of f(-x) or is less than that number by an even integer. If f(-x) has only one variation in sign, there is exactly one negative real zero.

25. Example 7 (like HW #43-56)
Find all roots of

26. You-Try #7 (like HW #43-56)
Find all zeros of

27. Assignment Complete one of the following assignments:
Page 302 #1-33 Every Other Odd, 43 OR
Page 302 #43-55 Odd
Chapter 2 Test next week
You may use a 3”x5” index card (both sides)