Real Zeros of Polynomial Functions

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 thanthe 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