# How do you find all the rational zeros of a polynomial function?

May 30, 2015

You can use the rational root theorem:

Given a polynomial of the form:

${a}_{0} {x}^{n} + {a}_{1} {x}^{n - 1} + \ldots + {a}_{n}$ with ${a}_{0} , \ldots , {a}_{n}$ integers,

all rational roots of the form $\frac{p}{q}$ written in lowest terms (i.e. with $p$ and $q$ having no common factor) will satisfy.

$p | {a}_{n}$ and $q | {a}_{0}$

That is $p$ is a divisor of the constant term and $q$ is a divisor of the coefficient of the highest order term.

This gives you a finite number of possible rational roots to try.

For example, the rational roots of

$6 {x}^{4} - 7 {x}^{3} + {x}^{2} - 7 x - 5 = 0$

must be of the form $\frac{p}{q}$ where $p$ is $\pm 1$ or $\pm 5$ and
$q$ is $1$, $2$, $3$ or $6$.

You can try substituting each of the possible combinations of $p$ and $q$ as $x = \frac{p}{q}$ into the polynomial to see if they work.

In fact the only rational roots it has are $- \frac{1}{2}$ and $\frac{5}{3}$.

Once you have found one root, you can divide the polynomial by the corresponding factor to simplify the problem.