## Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)

The maximum number of real zeros is $5$. The number of positive real zeros is $0$. The number of negative real zeros is either 3 or 1.
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 $5$. $f(x)=x^5+x^4+x^2+x+1$ has 0 variations in the sign. So, the number of positive real zeros is $0$. $f(−x)=-x^5+x^4+x^2-x+1$ has 3 variations in the sign. So, the number of negative real zeros is either 3 or 1.