Pre-CalculusChapter 2

Polynomial and Rational Functions

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Warm Up 2.5

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2.5 The Fundamental Theorem of Algebra

Objectives: Use the Fundamental Theorem of Algebra

to determine the number of zeros of a polynomial function.

Find all zeros of polynomial functions, including complex zeros.

Find conjugate pairs of complex zeros. Find zeros of polynomials by factoring.

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The Fundamental Theorem of Algebra

In the complex number system, every polynomial of degree n has exactly n zeros.

The zeros may be real or complex and they may repeat.

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Linear Factorization Theorem A polynomial of degree n has exactly n

linear factors.

The function f (x) can be written in the form

where c1, c2, …., cn are complex numbers. Note: The Fundamental Theorem of Algebra

and the Linear Factorization Theorem do not tell us how to find the zeros and factors. They just tell us that the zeros and factors exist.

nn cxcxcxaxf 21)(

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Example Justify that the third-degree polynomial

function

has exactly three zeros:

xxxf 4)( 3

ixixx 2,2,0

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Example Write as the

product of linear factors, and list all the zeros of f. Find the possible rational zeros of f.Use the graph to identify likely candidates.Use synthetic division to identify and

verify that one of these is actually a zero of f.

Factor the function completely and list the zeros.

8122)( 235 xxxxxf

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Example Write as the

product of linear factors, and list all the zeros of f. Find the possible rational zeros of f.Use the graph to identify likely candidates.Use synthetic division to identify and

verify that one of these is actually a zero of f.

Factor the function completely and list the zeros.

293911)( 23 xxxxf

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Conjugate Pairs Complex zeros occur in conjugate

pairs.

That is, if a + bi is a zero of a polynomial f, then a – bi is also a zero of f.

This is true only for functions that have real coefficients.

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Example Find a fourth-degree polynomial function with

real coefficients that has –1, –1, and 3i as zeros.

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Factors of a Polynomial The factors of a polynomial can be written as

linear complex factors.

Or, the factors can be written as linear and quadratic factors, when the quadratic factors have no real zeros.

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Example Write the polynomial as the product of linear

factors and quadratic factors in simplest (real) form. Then factor completely.

20)( 24 xxxf

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Example Find all zeros of the function given that 1 + 3i

is a zero of f.

60263)( 234 xxxxxf

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Homework 2.5

Worksheet 2.5# 1 – 9 odd# 25, 31, 33, 37, 39, 45, 49, 57