# How do you use the rational root theorem to list all possible rational roots?

Oct 25, 2015

See explanation...

#### Explanation:

Given a polynomial in the form:

$f \left(x\right) = {a}_{0} {x}^{n} + {a}_{1} {x}^{n - 1} + \ldots + {a}_{n - 1} x + {a}_{n}$

Any rational roots of $f \left(x\right) = 0$ are expressible as $\frac{p}{q}$ in lowest terms, where $p , q \in \mathbb{Z}$, $q \ne 0$, $p$ a divisor of ${a}_{n}$ and $q$ a divisor of ${a}_{0}$.

To list the possible rational roots, identify all of the possible integer factors of ${a}_{0}$ and ${a}_{n}$, and find all of the distinct fractions $\frac{p}{q}$ that result.

For example, suppose $f \left(x\right) = 6 {x}^{3} - 12 {x}^{2} + 5 x + 10$

Then:

${a}_{n} = 10$ has factors $\pm 1$, $\pm 2$, $\pm 5$ and $\pm 10$. (possible values of $p$)

${a}_{0} = 6$ has factors $\pm 1$, $\pm 2$, $\pm 3$ and $\pm 6$. (possible values of $q$)

Skip any combinations that have common factors (e.g. $\frac{10}{6}$) and list the resulting possible fractions:

$\pm \frac{1}{6}$, $\pm \frac{1}{3}$, $\pm \frac{1}{2}$, $\pm \frac{2}{3}$, $\pm \frac{5}{6}$, $\pm 1$, $\pm \frac{5}{3}$, $\pm 2$, $\pm \frac{5}{2}$, $\pm \frac{10}{3}$, $\pm 5$, $\pm 10$