# Polynomial remainder theorem

In algebra, the polynomial remainder theorem or little Bézout's theorem[1] is an application of Euclidean division of polynomials. It states that the remainder of the division of a polynomial $f(x)$ by a linear polynomial $x-a$ is equal to $f(a) .$ In particular, $x-a$ is a divisor of $f(x)$ if and only if $f(a)=0.$

## Examples

### Example 1

Let $f(x) = x^3 - 12x^2 - 42\,$. Polynomial division of $f(x)\,$ by $x-3\,$ gives the quotient $x^2 - 9x - 27\,$ and the remainder $-123\,$. Therefore, $f(3)=-123\,$.

### Example 2

Show that the polynomial remainder theorem holds for an arbitrary second degree polynomial $f(x) = ax^2 + bx + c$ by using algebraic manipulation:

\begin{align} \frac{f(x)}{{x - r}} &= \frac{{a{x^2} + bx + c}}{{x - r}} \\ &= \frac{{ax(x - r) + (b + ar)x + c}}{{x - r}} \\ &= ax + \frac{{(b + ar)(x - r) + c + r(b + ar)}}{{x - r}} \\ &= ax + b + ar + \frac{{c + r(b + ar)}}{{x - r}} \\ &= ax + b + ar + \frac{{a{r^2} + br + c}}{{x - r}} \end{align}

Multiplying both sides by (x − r) gives

$f(x) = ax^2 + bx + c = (ax + b + ar)(x - r) + {a{r^2} + br + c}$.

Since $R = ar^2 + br + c$ is our remainder, we have indeed shown that $f(r) = R$.

## Proof

The polynomial remainder theorem follows from the definition of polynomial long division; denoting the divisor, quotient and remainder by, respectively, $g(x)\,$, $q(x)\,$, and $r(x)\,$, polynomial long division gives a solution of the equation

$f(x)=q(x)g(x) + r(x)\,,$

where the degree of $r(x)\,$ is less than that of $g(x)\,$.

If we take $g(x) = x-a\,$ as the divisor, giving the degree of $r(x)\,$ as 0, i.e. $r(x) = r\,$:

$f(x)=q(x)(x-a) + r\,.$

Setting $x=a \!\,$ we obtain:

$f(a)=r\,.$

## Applications

The polynomial remainder theorem may be used to evaluate $f(a)\,$ by calculating the remainder, $r$. Although polynomial long division is more difficult than evaluating the function itself, synthetic division is computationally easier. Thus, the function may be more "cheaply" evaluated using synthetic division and the polynomial remainder theorem.

The factor theorem is another application of the remainder theorem: if the remainder is zero, then the linear divisor is a factor. Repeated application of the factor theorem may be used to factorize the polynomial.

## References

1. ^ Piotr Rudnicki (2004). "Little Bézout Theorem (Factor Theorem)". Formalized Mathematics 12 (1): 49–58.