# What are the rational zeros of a polynomial function?

May 30, 2016

See explanation...

#### Explanation:

A polynomial in a variable $x$ is a sum of finitely many terms, each of which takes the form ${a}_{k} {x}^{k}$ for some constant ${a}_{k}$ and non-negative integer $k$.

So some examples of typical polynomials might be:

${x}^{2} + 3 x - 4$

$3 {x}^{3} - \frac{5}{2} {x}^{2} + 7$

A polynomial function is a function wholse values are defined by a polynomial. For example:

$f \left(x\right) = {x}^{2} + 3 x - 4$

$g \left(x\right) = 3 {x}^{3} - \frac{5}{2} {x}^{2} + 7$

A zero of a polynomial $f \left(x\right)$ is a value of $x$ such that $f \left(x\right) = 0$.

For example, $x = - 4$ is a zero of $f \left(x\right) = {x}^{2} + 3 x - 4$.

A rational zero is a zero that is also a rational number, that is, it is expressible in the form $\frac{p}{q}$ for some integers $p , q$ with $q \ne 0$.

For example:

$h \left(x\right) = 2 {x}^{2} + x - 1$

has two rational zeros, $x = \frac{1}{2}$ and $x = - 1$

Note that any integer is a rational number since it can be expressed as a fraction with denominator $1$.