# What are necessary and sufficient conditions for a linear polynomial to be a factor of a given polynomial?

Jun 7, 2017

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#### Explanation:

Suppose you are given a polynomial:

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

and a potential linear factor:

$p x + q$

where all of ${a}_{n}$, ${a}_{n - 1}$,..., ${a}_{0}$, $p$ and $q$ are integers.

Then $p x + q$ can only be a factor of $f \left(x\right)$ if $p$ is a factor of ${a}_{n}$ and $q$ is a factor of ${a}_{0}$.

This immediately allows you to rule out many possibilities, but is not a sufficient condition.

What is sufficient is if $f \left(- \frac{q}{p}\right) = 0$.

In fact this last condition is both necessary and sufficient regardless of whether the coefficients are integers, rational, real or even complex numbers.