Remainder theorem and Factor theorem

Remainder theorem

The remainder of dividing a polynomial $p (x)$ by another one of the form $x - a$ , coincides with the value of $p (a)$ .

Notice that this kind of division satisfies the hypotheses of the Ruffini's rule.

Example

Calculate the remainder of the division $\frac{p (x)}{q (x)}$ , where $p (x) = x^{4} + 3 x^{2} - x + 4$ and $q (x) = x + 2$ .

We apply the remainder theorem. Notice that, in this case $a = - 2$ . $p (- 2) = (- 2)^{4} + 3 \cdot (- 2)^{2} - (- 2) + 4 = 16 + 3 \cdot 4 + 2 + 4 = 34$

To verify it we use Ruffini:

	$1$	$0$	$3$	$- 1$	$4$
$- 2$		$- 2$	$4$	$- 14$	$30$
	$1$	$- 2$	$7$	$- 15$	$34$

And, it is the same as the previous solution.

Example

Calculate the remainder of the division $\frac{p (x)}{q (x)}$ , where $p (x) = x^{5} - 2 x^{2} + x + 3$ and $q (x) = x + 1$ .

We apply the remainder theorem. Notice that, in this case $a = - 1$ . $p (- 1) = (- 1)^{5} - 2 \cdot (- 1)^{2} + (- 1) + 3 = - 1 - 2 - 1 + 3 = - 1$

To verify it we use Ruffini:

	$1$	$0$	$0$	$- 2$	$1$	$3$
$- 1$		$- 1$	$1$	$- 1$	$3$	$- 4$
	$1$	$- 1$	$1$	$- 3$	$4$	$- 1$

And it is the same than the previous solution.

Factor theorem

Its statement is the following one:

A polynomial $p (x)$ is divisible by another of the form $x - a$ if, and only if, $p (a) = 0$ . In this case, we will say that $a$ is a root or zero of the polynomial $p (x)$ .

Example

Calculate the remainder of the division $\frac{p (x)}{q (x)}$ , where $p (x) = x^{5} + 2 x^{4} - 3 x^{3} + x^{2} - 1$ and $q (x) = x - 1$ .

We apply the remainder theorem $p (1) = 1^{5} + 2 \cdot 1^{4} - 3 \cdot 1^{3} + 1^{2} - 1 = 0$

We verify the result using Ruffini:

	$1$	$2$	$- 3$	$1$	$0$	$- 1$
$1$		$1$	$3$	$0$	$1$	$2$
	$1$	$3$	$0$	$1$	$1$	$0$

Indeed, the remainder is $0$ . And so, according to the factor theorem, the division of $p (x)$ by $q (x)$ is exact.