Answer
$f(x)$ has $4$ or $2$ positive zeros. and $f(x)$ has $2$ or $0$ negative zeros
Work Step by Step
Descartes’ Rule of Signs
Let $f$ denote a polynomial function written in standard form.
The number of positive real zeros of $f$ either equals the number of variations in the sign of the nonzero coefficients of $f(x)$ or else equals that number less an even integer.
The number of negative real zeros of $f$ either equals the number of variations in the sign of the nonzero coefficients of $f(-x)$ or else equals that number less an even integer
Therefore, $f(x)=12x^8-x^7+8x^4-2x^3+x+3,$ has $4$ or $2$ positive zeros. and $f(-x)=12x^8+x^7+8x^4+2x^3-x+3$ has $2$ or $0$ negative zeros
