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

The maximum number of real zeros is $6$. The number of positive real zeros is either 2 or 0. The number of negative real zeros is either 2 or 0.
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 $6$, so the maximum number of real zeros is $6$. $f(x)=2x^6-3x^2-x+1$ has 2 variations in the sign. So, the number of positive real zeros is either 2 or 0. $f(−x)=2x^6-3x^2+x+1$ has 2 variations in the sign. So, the number of negative real zeros is either 2 or 0.