Download presentation

Presentation is loading. Please wait.

Published byAntoine Dingley Modified over 2 years ago

1
**2.5 complex 0’s & the fundamental theorem of algebra**

2
**Complex Conjugate Zeros**

Suppose that f(x) is a polynomial function with real coefficients. If a and b are real numbers with b ≠ 0 and a + bi is a zero of f(x), then its complex conjugate a - bi is also a zero of f(x). Example: find the complex conjugate 4 – 3i 8 + 2i 3 + √3 i

3
**Factoring a complex polynomial function**

Use the Rational Root theorem Examples: find all zeros f(x) = 3x4 + 8x3 + 6x2 + 3x – 2 f(x) = x3 – 10x2 + 44x – 69

4
**Fundamental theorem of algebra**

A polynomial of degree n has n complex roots (real and non-real). Some of these may be repeated. In other words: if you can solve the polynomial the polynomial has 0’s For example: if k is a zero of a polynomial if and only if x – k is a factor of the polynomial

5
**How to use the fundamental theorem of algebra**

2 parts: 1) find all roots, how you do this depends on if you are given a real root or a complex root a) given a real root: do synthetic division, rewrite your answer, find the remaining zeros by doing quadratic formula (these will be complex numbers),write your answer in set notation.) b) given a complex root: find the conjugate, write both as (x – (a + bi))(x – (a – bi)) and foil to get rid of the i’s, take that quadratic and the original problem and find the remaining roots by doing long division. If need be factor or use quadratic formula to find the remaining roots or set equal to 0 and solve, write your answer in set notation.

6
**Cont’d 2) given the roots find the polynomial function**

write your roots as (x - #)(x - #),etc. If given a complex root or a radical, DO NOT forget about it’s conjugate. Multiply the parenthesis together either by FOIL or distributing. Write your final answer as f(x) = …

7
**Use the fundamental theorem of algebra**

Use the fundamental theorem of algebra to find all zeros Examples: f(x) = x3 – 3x2 + 9x + 13; 2 – 3i f(x) = x3 + 6x2 + 21x + 26; -2

8
**Finding a polynomial given zeros**

**if there is an i or radical in 1 parenthesis you must have the conjugate in another Examples: write a polynomial with minimal degree in standard form with real coefficients whose zeros include those listed. Use the fundamental theorem of algebra -4, 1 – i, and 1 + i -2, 1- i -2- i, 1 + 3i

9
Examples cont’d 4. -1 (multiplicity 3), 3 (multiplicity 1) (x – 1)3(x – 3) = (x – 1)(x – 1)(x – 1)(x – 3) (multiplicity 2), -2 – i (multiplicity 1)

10
**Fundamental Polynomial connections in the complex case**

The following statements about a polynomial function f are equivalent if k is a complex number: 1. x = k is a solution (or root) of the equation f(x) = 0 2. k is a zero of the function f 3. x – k is a factor of f(x)

11
**Write the function in std**

Write the function in std. form & identify the 0’s and x-intercepts of the graph Examples: f(x) = (x – 2i)(x + 2i) f(x) = (x – 5)(x - √2 i)(x + √2 i) f(x) = (x – 1)(x – 1)(x + 2i)(x – 2i)

Similar presentations

OK

Lesson 3.4 – Zeros of Polynomial Functions Rational Zero Theorem

Lesson 3.4 – Zeros of Polynomial Functions Rational Zero Theorem

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Show ppt on drainage pattern of indian rivers Ppt on power grid failure solar Ppt on what is critical whiteness theory Ppt on power diode characteristics Ppt on mammals and egg laying animals wikipedia Ppt on coalition government in canada Ppt on cryogenic engine Free ppt on personality development for students Ppt on preparation and properties of alcohols and phenols Ppt on cross-sectional study vs cohort study