# What is the rational zeros theorem?

Aug 7, 2018

See explanation...

#### Explanation:

The rational zeros theorem can be stated:

Given a polynomial in a single variable with integer coefficients:

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

with ${a}_{n} \ne 0$ and ${a}_{0} \ne 0$, any rational zeros of that polynomial are expressible in the form $\frac{p}{q}$ for integers $p , q$ with $p$ a divisor of the constant term ${a}_{0}$ and $q$ a divisor of the coefficient ${a}_{n}$ of the leading term.

Interestingly, this also holds if we replace "integers" with the element of any integral domain. For example it works with Gaussian integers - that is numbers of the form $a + b i$ where $a , b \in \mathbb{Z}$ and $i$ is the imaginary unit.