# Timeless Theorems of Mathematics/Polynomial Remainder Theorem

The **Polynomial Remainder Theorem** is an application of Euclidean
division of polynomials. It is one of the most fundamental and popular
theorems of Algebra. It states that the remainder of the division of a
polynomial $P(x)$ by a linear polynomial $(x - r)$ is equal to $P(r)$.

## Examples

### Example 1

Show that the remainder of the division of a polynomial
$f(x) = x^2 - 2x + 2$ by a linear polynomial $(x - 1)$ is equal to
$f(1)$. **Solution :** Divide $f(x) = x^2 - 2x + 2$ by $(x - 1)$ like
the following one.

``` cpp
x - 1 ) x^2 - 2x + 2 ( x - 1
        x^2 - x
        ------------
            - x + 2
            - x + 1
        ------------
                  1
```

As, $f(1) = 1^2 - 2(1) + 2 = 1$, thus the remainder is equal to $f(1)$.

### Example 2

Show that the remainder of the division of a polynomial
$F(x) = ax^2 + bx + c$ by a linear polynomial $(x - m)$ is equal to
$F(m)$. **Solution :** Divide $F(x) = ax^2 + bx + c$ by $(x - m)$ like
the following one.

``` cpp
x-m ) ax^2+bx+c ( ax+am+b
      ax^2-amx
      ------------------
           amx+bx+c
           amx     -am^2
      ------------------
               bx+c+am^2
               bx-bm
      ------------------
               am^2+bm+c
```

As, $f(m) = am^2 + bm + c$, thus the remainder $am^2 + bm + c$ is equal
to $f(m)$.

## Proof

### Proposition

If $P(x)$ is a polynomial of a positive degree and $a$ is any definite
number, the remainder of the division of $P(x)$ by $(x - a)$ will be
$P(a)$

### Proof

The remainder of the division of a polynomial of a positive degree
$P(x)$ by $(x - a)$ is either 0 or a non-zero constant. Let the
remainder is $R$ and the quotient is $Q(x)$. Then, for every value of
$x$,

```{=html}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
```
$P(x) = (x - a)$·$Q(x) + R .... (i)$

```{=html}
</div>
```
Putting $x = a$ in the equation $(i)$, we get $P(a) = 0$ ·
$Q(a) + R = R$. Thus, the remainder of $P(x)$÷$(x - a)$ is equal to
$P(a)$.

