Download presentation

Presentation is loading. Please wait.

Published byBerenice Jacobs Modified over 6 years ago

1
Zeros of Polynomial Functions Section 2.5 Page 312

2
Review Factor Theorem: If x – c is a factor of f(x), then f(c) = 0 Example: x – 3 is a zero of f(x) = 2x 3 – 3x 2 – 11x + 6

3
Information about The Rational Zero Theorem Use to find possible rational zeros Provides a list of possible rational zeros of a polynomial function Not every number will be a zero

4
The Rational Zero Theorem If f(x) has integer coefficients and p/q (where p/q is reduced to lowest terms) is a rational zero of f, the p is a factor of the constant term a 0 and q is a factor of the leading coefficient a n Possible zeros = factors of a 0 = p factors of a n q

5
Example 1 List all possible zeros of f(x) = -x 4 + 3x 2 + 4

6
Example 2 List all possible zeros of f(x) = 15x 3 + 14x 2 – 3x – 2

7
Finding zeros Use the Rational Zero Theorem and trial & error to find a rational zero Once the polynomial is reduced to a quadratic then use factoring or the quadratic formula to find the remaining zeros.

8
Example 3 Find the zeros of f(x) = x 3 + 2x 2 – 5x – 6

9
Example 4 Find all the zeros of f(x) = x 3 + 7x 2 + 11x – 3

10
Example 5 Solve: x 4 – 6x 2 – 8x + 24 = 0

11
Practice List all possible rational zeros 1. f(x) = 4x 5 + 12x 4 – x – 3 Find all zeros 2. f(x) = x 3 + 8x 2 + 11x – 20 3. f(x) = x 3 + x 2 – 5x – 2 Solve 4. x 4 – 6x 3 + 22x 2 – 30x + 13 = 0

12
Properties of Polynomial Equations 1. If a polynomial equation is of degree, n, then counting multiple roots separately, the equation has n roots. 2. If a + bi is a root of a polynomial equation with real coefficients (b ≠ 0), then the complex imaginary number a – bi is also a root. Complex imaginary roots, if they exist, occur in conjugate pairs. If 3i is a root, then –3i is also a root If 2 – 5i is a root, then 2 + 5i is also a root

13
The Fundamental Theorem of Algebra If f(x) is a polynomial of degree n, where n ≥ 1, then the equation f(x) = 0 has at least one complex root.

14
Example 6 Find a 4 th degree polynomial function f(x) with real coefficients that has -2, 2, and i as zeros and such that f(3) = -150

15
Descartes’s Rule of Signs 1.The number of positive real zeros of f is either a. The same as the number of sign changes of f(x) OR b. Less than the number of sign changes by an even integer Note: if f(x) has only one sign change, then f has only one positive real zero 2. The number of negative real zeros of f is either a. The same as the number of sign changes of f(-x) OR b. Less than the number of sign changes by an even integer Note: if f(-x) has only one sign change, then f has only one negative real zero

16
Review Table on page 320 Negative Real Zeros f(-x) = -3x 7 + 2x 5 – x 4 + 7x 2 – x – 3 f(-x) = -4x 5 + 2x 4 – 3x 2 – x + 5 f(-x) = -7x 6 - 5x 4 – x + 9

17
Example 7 Determine the possible numbers of positive and negative real zeros of f(x) = x 3 + 2x 2 + 5x + 4

18
Practice Find a third-degree polynomial function with real coefficients that has -3 and i as zeros such that f (1) = 8 Determine the possible numbers of positive and negative real zeros of f(x) = x 4 - 14x 3 + 71x 2 – 154x + 120

Similar presentations

© 2022 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google