# How do you use the rational zero theorem to list all possible rational zeros for the given function f(x)=x^3-14x^2+13x-14?

Feb 18, 2015

Hello,

Let $P = {a}_{n} {X}^{n} + {a}_{n - 1} {X}^{n - 1} + \setminus \ldots + {a}_{1} X + {a}_{0}$ a polynom with ${a}_{0} , \setminus \ldots , {a}_{n}$ integers. Suppose that ${a}_{n} \setminus \ne 0$ and ${a}_{0} \setminus \ne 0$.

If irreductible fraction $\setminus \frac{p}{q}$ is a root of $P$, then $p$ is a factor of ${a}_{0}$ and $q$ is a factor of ${a}_{n}$.

In your example, the factors of ${a}_{0} = - 14$ are
$- 1 , 1 , - 2 , 2 - 7 , 7 , - 14 , 14$,
and the factors of ${a}_{n} = 1$ are just $- 1 , 1$.

Therefore, you have some candidates for the rational roots :
$- 1 , 1 , - 2 , 2 - 7 , 7 , - 14 , 14$.

Remark. If ${a}_{n} = 1$, the rational roots are necessarily integers !

You can check that no one is solution !