Listing All Possible Rational Zeros

The only possible rational zeros of $f(x)$ are the quotients of the factors of the last term, –4, and the factors of the leading coefficient, 2.

The constant term is –4; the factors of –4 are $p=\mathrm{\pm 1},\mathrm{\pm 2},\mathrm{\pm 4.}$

The leading coefficient is 2; the factors of 2 are $q=\mathrm{\pm 1},\mathrm{\pm 2.}$

If any of the four real zeros are rational zeros, then they will be of one of the following factors of –4 divided by one of the factors of 2.

Note that $\frac{2}{2}=1$ and $\frac{4}{2}=2,$ which have already been listed. So we can shorten our list.