College Algebra (10th Edition)

Published by Pearson
ISBN 10: 0321979478
ISBN 13: 978-0-32197-947-6

Chapter 5 - Section 5.5 - The Real Zeros of a Polynomial Function - 5.5 Assess Your Understanding: 30

Answer

The maximum number of real zeros is $5$ The number of positive real zeros are $5$ or $3$ or $1$ The number of negative real zeros is $0$

Work Step by Step

The number of zeros of a polynomial can’t be greater than its degree. $1)$ The number of positive real zeros of $f(x) $ either equals the number of variations in the sign of the nonzero coefficients of $f(x)$ or equals that number minus an even integer $2)$ The number of negative real zeros of $f(x)$ either equals the number of variations in the sign of the nonzero coefficients of $f(-x) $ or equals that number minus an even integer. So the maximum number of real zeros here is $5$ Since $$f\left( x\right) =x^{5}-x^{4}+x^3-x^{2}+x-1 $$ has $5$ variations The number of positive real zeros is $5$ Or $3$ Or $1$ Since $$ f\left( -x\right) = -x^{5}-x^{4}-x^3-x^{2}-x-1 $$ Has $0$ variations The number of negative real zeros is $0$
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