# Chapter 5 - Section 5.5 - The Real Zeros of a Polynomial Function - 5.5 Assess Your Understanding - Page 387: 27

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

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