College Algebra (10th Edition)

Published by Pearson

Chapter 5 - Section 5.5 - The Real Zeros of a Polynomial Function - 5.5 Assess Your Understanding - Page 387: 21

Answer

Maximum number of real zeros: $7$ The number of positive real zeros is either $3$ or $1$ The number of negative real zeros is either $2$ or $0$

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 zero here is $7$. Since: $$f\left( x\right) =-4x^{7}+x^{3}-x^{2}+2$$ has $3$ variations The number of positive real zeros is either $3$ or $1$ Since $$f\left( -x\right) =4x^{7}-x^{3}-x^{2}+2$$ Has $2$ variations The number of negative real zeros is either $2$ or $0$.

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