1. Real Zeros of Polynomial Functions Lesson 4.4

2. Division of Polynomials Can be done manually See Example 2, pg 253 Calculator can also do division Use propFrac( ) function

3. Division Algorithm For any polynomial f(x) with degree n ≥ 0 There exists a unique polynomial q(x) and a number r Such that f(x) = (x – k) q(x) + r The degree of q(x) is one less than the degree of f(x) The number r is called the remainder

4. Remainder Theorem If a polynomial f(x) is divided by x – k The remainder is f(k)

5. Factor Theorem When a polynomial division results in a zero remainder The divisor is a factor f(x) = (x – k) q(x) + 0 This would mean that f(k) = 0 That is … k is a zero of the function

6. Completely Factored Form When a polynomial is completely factored, we know all the roots

7. Zeros of Odd Multiplicity Given Zeros of -1 and 3 have odd multiplicity The graph of f(x) crosses the x-axis

8. Zeros of Even Multiplicity Given Zeros of -1 and 3 have even multiplicity The graph of f(x) intersects but does not cross the x-axis

9. Try It Out Consider the following functions Predict which will have zeros where The graph intersects only The graph crosses

10. From Graph to Formula If you are given the graph of a polynomial, can the formula be determined? Given the graph below: What are the zeros? What is a possible set of factors? Note the double zero

11. From Graph to Formula Try graphing the results... does this give the graph seen above (if y tic-marks are in units of 5 and the window is -30 < y < 30) The graph of f(x) = (x - 3) 2 (x+ 5) will not go through the point (-3,7.2) We must determine the coefficient that is the vertical stretch/compression factor... f(x) = k * (x - 3)2(x + 5)... How?? Use the known point (-3, 7.2) 7.2 = f(-3) Solve for k

12. Assignment Lesson 4.4 Page 296 Exercises 1 – 53 EOO 73 – 93 EOO