College Algebra 7th Edition

Published by Brooks Cole
ISBN 10: 1305115546
ISBN 13: 978-1-30511-554-5

Chapter 3, Polynomial and Rational Functions - Section 3.4 - Real Zeros of Polynomials - 3.4 Exercises - Page 320: 65

Answer

$1$ positive root, $1$ negative root $2$ real roots

Work Step by Step

Descartes' Rule of Signs states that the possible number of the positive roots of a polynomial is equal to the number of sign changes in the coefficients or less than the number of sign changes by a multiple of $2$ AND the possible number of negative roots of a polynomial is equal to the number of sign changes or less than the total number of sign changes by a multiple of $2$ after substituting $−x$ for $x$. The substitution has the effect of negating all of the odd-power terms in the polynomial. Therefore, $P(x)=2x^6+5x^4-x^3-5x-1$, has $1$ sign change, between $5x^4$ and $-x^3$. Therefore, it has $1$ positive root. $P(-x)=2x^6+5x^4+x^3+5x-1$, has $1$ sign change, between $5x$ and $-1$. Therefore, it has $1$ negative root. Therefore, It has a total of $2$ real roots.
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