# 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 $69$ - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**remainder theorem**:**1.**