## Algebra 2 (1st Edition)

Published by McDougal Littell

# Chapter 5 Polynomials and Polynomial Functions - 5.7 Apply the Fundamental Theorem of Algebra - 5.7 Exercises - Skill Practice - Page 384: 40

#### Answer

Positive real zeros = 1 or 3 Negative real zeros= 1 or 3 Imaginary zeros = 4, 2 or 0

#### Work Step by Step

In the given polynomial of degree $6$ , there are $6$ zeros. Here, we have $g(x)=x^6+x^5-3x^4+x^3+5x^2+9x-18$ By Descartes' Rule, we have $3$ sign changes, thus we have either $3$ or $1$ positive real roots. Now, $g(-x)=x^6-x^5-3x^4-x^3+5x^2-9x-18$ The above polynomial shows that there are $3$ sign changes. By Descartes' Rule, we have $3$ or $1$ negative real roots. Thus, the other two roots must be imaginary. Hence, Positive real zeros = 1 or 3 Negative real zeros= 1 or 3 Imaginary zeros = 4, 2 or 0

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