Answer
$7$.
3 or 1 positive real zeros.
2 or 0 negative real zeros.
Work Step by Step
Step 1. Based on the given function, we have degree$n=7$, thus the maximum number of real zeros that the polynomial function may have is $7$.
Step 2. As there are 3 sign changes in $f(x)$, based on the Descartes’ Rule of Signs, there may be 3 or 1 positive real zeros.
Step 3. $f(-x)=4x^7-x^3-x^2+2$ and there are 2 sign changes in $f(-x)$, thus there may be 2 or 0 negative real zeros.