Chapter 5 - Section 5.5 - The Real Zeros of a Polynomial Function - 5.5 Assess Your Understanding - Page 387: 31

The maximum number of real zeros is $6$ The number of positive real zeros is $1$ The number of negative real zeros is $1$

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 $6$ Since $$f\left( x\right) =x^{6}-1 $$ has $1$ variation The number of positive real zeros is $1$. Since $$ f\left( -x\right) = x^{6}-1 $$ Has $1$ variation The number of negative real zeros is $1$.