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 $0$
