2.2 POLYNOMIAL FUNCTIONS OF HIGHER DEGREE Copyright © Cengage Learning. All rights reserved.

2 Find and use zeros of polynomial functions as sketching aids. Write a polynomial function given its degree and zeros What You Should Learn

3 Zeros of Polynomial Functions

4 It can be shown that for a polynomial function f of degree n, the following statements are true. 1. The function f has, at most, n real zeros. 2. The graph of f has, at most, n – 1 turning points. (Turning points, also called relative minima or relative maxima, are points at which the graph changes from increasing to decreasing or vice versa.) Finding the zeros of polynomial functions is one of the most important problems in algebra.

5 Zeros of Polynomial Functions

6 Example 1 – Finding the Zeros of a Polynomial Function Find all real zeros of f (x) = –2x 4 + 2x 2. Then determine the number of turning points of the graph of the function. Solution: To find the real zeros of the function, set f (x) equal to zero and solve for x. –2x 4 + 2x 2 = 0 Set f (x) equal to 0.

7 Example 1 – Solution –2x 2 (x 2 – 1) = 0 –2x 2 (x – 1)(x + 1) = 0 So, the real zeros are x = 0, x = 1, and x = –1. Because the function is a fourth-degree polynomial, the graph of f can have at most 4 – 1 = 3 turning points. Remove common monomial factor. Factor completely. cont’d

8 Zeros of Polynomial Functions In Example 1, note that because the exponent is greater than 1, the factor –2x 2 yields the repeated zero x = 0. Because the exponent is even, the graph touches the x-axis at x = 0, as shown in Figure Figure 3.20

9 Zeros of Polynomial Functions A polynomial function is written in standard form if its terms are written in descending order of exponents from left to right. Before applying the Leading Coefficient Test to a polynomial function, it is a good idea to check that the polynomial function is written in standard form.

10 Example: Writing Polynomials Find a polynomial that has the given zeros 0, 2, and 5.

11 Example: Writing Polynomials Find a polynomial of degree 3 that has the given zeros -2, 4, and 7.