# Little Bézout Theorem

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## Theorem

Let $\map {P_n} x$ be a polynomial of degree $n$ in $x$.

Let $a$ be a constant.

Then the remainder of $\map {P_n} x$ when divided by $x - a$ is equal to $\map {P_n} a$.

## Proof

By the process of Polynomial Long Division, we can express $\map {P_n} x$ as:

- $(1): \quad \map {P_n} x = \paren {x - a} \map {Q_{n - 1} } x + R$

where:

- $\map {Q_{n - 1} } x$ is a polynomial in $x$ of degree $n - 1$

It follows that, by setting $x = a$ in $(1)$, we get $\map {P_n} a = R$.

Hence the result.

$\blacksquare$

## Also known as

This theorem is sometimes referred to as the Polynomial Remainder Theorem.

Some sources call it merely the Remainder Theorem, but there is more than one theorem so named.

## Source of Name

This entry was named for Étienne Bézout.

## Sources

- 1960: Margaret M. Gow:
*A Course in Pure Mathematics*... (previous) ... (next): Chapter $1$: Polynomials; The Remainder and Factor Theorems; Undetermined Coefficients; Partial Fractions: $1.2$. The remainder and factor theorems - 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $8$: Field Extensions: $\S 37$. Roots of Polynomials: Theorem $69$