# 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.