1. 3.3 (1) Zeros of Polynomials Multiplicities, zeros, and factors, Oh my

2. PSAT Review Let’s review from sample test #4. We’ll look at #31, 37, 38. Others?

3. POD Factor into linear factors and find the zeros. Graph them to confirm your zeros. 1.6x 3 - 2x 2 - 6x + 2 2.5x 3 - 30x 2 - 65x What do you notice about the number of linear factors and the number of zeros?

4. POD Factor into linear factors and find the zeros. Graph them to confirm your zeros. 1.6x 3 - 2x 2 - 6x + 2 = 2(3x 3 - x 2 - 3x + 1) = 2(x 2 (3x – 1) – (3x – 1) = 2(x 2 – 1)(3x-1) = 2(x + 1)(x – 1)(3x – 1) 2.5x 3 - 30x 2 - 65x = 5x(x 2 – 6x – 13) = 5x(x – (3 + √22))(x – (3 – √22)) What do you notice about the number of linear factors and the number of zeros?

5. The relationship between zeros and factors If we include real and complex zeros, and consider multiplicities of zeros, there are the same number of zeros as there are linear factors. How does this relate to the degree of the polynomial? What are other names for “zeros?” x-intercepts are what type of zero? Does this mean every linear factor represents an x-intercept? What sorts of factors do we get if we limit them to real numbers?

6. Use it We’ve seen the match up between linear factors and zeros in the POD. Now, find zeros of f(x) and g(x) with algebra and by graphing. f(x) = x 4 - 3x 3 +2x 2 g(x) = x 5 - 4x 4 +13x 3

7. Use it We’ve seen the match up between linear factors and zeros in the POD. Now, find zeros of f(x) and g(x) with algebra and by graphing. f(x) = x 4 - 3x 3 +2x 2 How do they match up here?

8. Use it We’ve seen the match up between linear factors and zeros in the POD. Now, find zeros of f(x) and g(x) with algebra and by graphing. g(x) = x 5 - 4x 4 +13x 3 How do they match up here?

9. Use it Find f(x) with zeros at x = -5, 2, and 4. (How many of these could we come up with? What would they look like? How many could be third degree?) Now, add to those zeros that f(3) = -24. What does the equation become?

10. Use it Find the zeros and their multiplicities of 1.f(x) = x 2 (3x + 2)(2x - 5) 3 2.g(x) = (x 2 + x - 12) 3 (x 2 - 9) What is the degree of each of these polynomials? How many linear factors does each have? How many zeros does each have? Are they all real? How many time does the graph cross the x- axis?

11. Use it Find the zeros and their multiplicities of 1.f(x) = x 2 (3x + 2)(2x - 5) 3 = xx(3x + 2)(2x – 5)(2x – 5)(2x – 5) ZeroMultiplicityDegree of six 0 2Six linear factors -2/3 1Three zeros– all real 5/2 3Crosses the x-axis 3 times 2.g(x) = (x 2 + x - 12) 3 (x 2 - 9) = (x + 4) 3 (x – 3) 3 (x – 3)(x + 3) ZeroMultiplicityDegree of eight -4 3Eight linear factors 3 4Three zeros– all real -3 1Crosses the x-axis 3 times

12. Use it Create your own. Write a polynomial function with an odd number of real roots and a pair of imaginary roots. Give it with linear factors. Give it with real number factors.

13. Graphs of multiplicities– review On calculators, graph f(x) = x - 1 g(x) = (x - 1) 3 h(x) = (x - 1) 5 Next, graphf(x) = (x - 1) 2 g(x) = (x - 1) 4 h(x) = (x - 1) 6 What do you notice about the exponents and the graphs?

14. Graphs of multiplicities—review f(x) = (x – 1) g(x) = (x - 1) 3 h(x) = (x - 1) 5 f(x) = (x - 1) 2 g(x) = (x - 1) 4 h(x) = (x - 1) 6

15. Graphs of multiplicities– review In a graph of f(x) = (x - c) m, if c is a real number, the graph will cross the x-axis at c if m is odd. the graph will touch the x-axis at c, but not cross it, if m is even.