# Condition for Linear Divisor of Polynomial

## Theorem

Let $\map P x$ be a polynomial in $x$.

Let $a$ be a constant.

Then $x - a$ is a divisor of $\map P x$ if and only if $a$ is a root of $P$.

## Proof

From the Little Bézout Theorem, the remainder of $\map P x$ when divided by $x - a$ is equal to $\map P a$.

### Sufficient Condition

Let $x - a$ be a divisor of $\map P x$.

From the Little Bézout Theorem, the remainder of $\map P x$ when divided by $x - a$ is equal to $\map P a$.

By definition of divisor, the remainder of $\map P x$ when divided by $x - a$ equals $0$.

That is:

- $\map P a = 0$

It follows by definition that $a$ is a root of $P$.

$\Box$

### Necessary Condition

Let $a$ be a root of $P$.

From the Little Bézout Theorem, and by Polynomial Long Division, we have:

- $\map P x = \paren {x - a} \map Q x + \map P a$

where $\map Q x$ is a polynomial in $x$ of degree one less than $\map P x$.

By definition of root of polynomial:

- $\map P a = 0$

So we have:

- $\map P x = \paren {x - a} \map Q x$

and the result follows by definition of divisor.

$\blacksquare$

## Sources

- 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $8$: Field Extensions: $\S 37$. Roots of Polynomials: Theorem $70$