Answer
The maximum number of real zeros is $5$.
The number of positive real zeros is 5 or, 3 or, 1.
The number of negative real zeros is $0$.
Work Step by Step
We should remember that the maximum number of zeros of a polynomial function $f(x)$ cannot be greater than its degree.
We will consider Descartes' Rule of Signs for explaining this solution:
(1) The number of negative real zeros of a polynomial function $f(x)$ is either equal to the number of variations in the sign of the non-zero coefficients of $f(−x)$ or that number minus an even integer.
(2) The number of positive real zeros of a polynomial function $f(x)$ is either equal to the number of variations in the sign of the non-zero coefficients of $f(x)$ or that number minus an even integer.
We can notice from the given polynomial function that the highest degree is $5$, so the maximum number of real zeros is $5$.
$f(x)=x^5-x^4+x^3-x^2+x-1$ has 5 variations in the sign. So, the number of positive real zeros is 5 or, 3 or, 1.
$f(−x)=-x^5-x^4-x^3-x^2-x-1$ has 0 variations in the sign. So, the number of negative real zeros is $0$.
