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.

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

We apply the remainder theorem. Notice that, in this case $a=-2$. $$p(-2)=(-2)^4+3\cdot(-2)^2-(-2)+4=16+3\cdot4+2+4=34$$

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.

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

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

We apply the remainder theorem $$p(1)=1^5+2\cdot1^4-3\cdot1^3+1^2-1=0$$

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.