1. 2.7 Apply the Fundamental Theorem of Algebra Polynomials Quiz: Tomorrow (over factoring and Long/Synthetic Division) Polynomials Test: Friday

2. Vocabulary  When a factor of a polynomial appears more than once, you can count its solution more than once. The repeated factor produces a repeated solution.  If f(x) is a polynomial of degree n where n > 0, then the equation f(x) = 0 has exactly n solutions provided each solution repeated twice is counted as 2 solutions, each solution repeated three times is counted as 3 solutions, and so on.

3. Vocabulary  Complex Conjugates Theorem: If f is a polynomial function with real coefficients, and a + bi is an imaginary zero of f, then a – bi is also a zero of f.

4. Vocabulary:  Irrational Conjugates Theorem: Suppose f is a polynomial function with rational coefficients, and a and b are rational numbers such that is irrational. If is a zero of f, then is also a zero of f.

5. Example 1:  Find all zeros of f(x) = x 5 – x 4 – 7x 3 + 11x 2 + 16x – 20 Step 1: Find all the rational zeros of f. -2 is a zero repeated twice and 1 is a zero, so… (x + 2)(x + 2)(x – 1) **(x + 2)(x + 2)(x – 1) does not multiply out to give us the original function**

6. Example 1: Continued Step 2: Divide f by its known factors x + 2, x + 2, and x - 1 -2 1 -1 -7 11 16 - 20 Step 3: Find the complex zeros. Now, divide what you got by the next factor. And so on until you have divided with all factors

7. You Try:  Find all the zeros of the function f(x) = x 4 – 4x 3 + 5x 2 – 2x - 12

8. Example 2:  Write a polynomial function of least degree that has rational coefficients, a leading coefficient of 1, and 4 and as zeros.

9. You Try:  Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros. i, 4

10. Homework:  Study for Polynomial Quiz tomorrow I will post the answers to the Factoring WS online. Look over Long/Synthetic Division