Warm-Up: January 9, 2012

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)