list all possible rational zeros of f.
f(x)=3x^4 + 4x^3 -5x^2 -8
WRONG!, that would make $\displaystyle \sqrt{2}$ a rational zero of $\displaystyle x^2-2$
A rational number is a number that can be written as the ratio of two integers.
This question is meant to be addressed using the rational root theorem, which tells you that any rational root of a polynomial equation with integer coefficients is the ratio of a factor of the constant term to a factor of the coefficient of the highest power occuring.
So for:
$\displaystyle 3x^4 + 4x^3 -5x^2 -8 = 0$
the rational roots are amoung $\displaystyle \pm1,\ \pm 2,\ \pm 4,\ \pm 8,\ \pm1/3,\ \pm 2/3,\ \pm 4/3,\ \pm 8/3$
You try each of these to find which are actual roots.
CB