Remainder and Factor Theorem

The Remainder Theorem:

To divide two polynomials, say $P\left(x\right)$P(x) divided by $D\left(x\right)$D(x), we need the degree of the divisor polynomial $D\left(x\right)$D(x) to be less than or equal to the degree of the dividend polynomial $P\left(x\right)$P(x). An earlier entry on polynomial division can be found here.

As an example suppose we divide $P\left(x\right)=x^2-5x+6$P(x)=x2−5x+6 by the monic polynomial $D\left(x\right)=x-1$D(x)=x−1. We proceed in a manner similar to long division given as follows:

Thus we state:

$\frac{P\left(x\right)}{D\left(x\right)}=\frac{x^2-5x+6}{x-1}=\left(x-4\right)+\frac{2}{x-1}$P(x)D(x)​=x2−5x+6x−1​=(x−4)+2x−1​

We can express the result slightly differently by multiplying both sides by the divisor so that:

$P\left(x\right)=x^2-5x+6=\left(x-1\right)\left(x-4\right)+2$P(x)=x2−5x+6=(x−1)(x−4)+2

Stating the result like this is known as the division transformation.

In general, dividing $P\left(x\right)$P(x) by $\left(x-a\right)$(x−a) will always produce a result that looks like:

$P\left(x\right)=\left(x-a\right)Q\left(x\right)+R$P(x)=(x−a)Q(x)+R

Here, $Q\left(x\right)$Q(x) is the quotient polynomial of one less degree than $P\left(x\right)$P(x) and $R$R is the remainder.

This last equation holds the key to the remainder theorem. Because the polynomial holds true for all values of $x$x, by putting $x=a$x=a into this general result we see that:

$P\left(a\right)=\left(a-a\right)Q\left(a\right)+R=R$P(a)=(a−a)Q(a)+R=R

This means that substituting $x=a$x=a into $P\left(x\right)$P(x) before dividing will reveal the remainder. It's a little mathematical magic! We can actually know the remainder even before the division by $\left(x-a\right)$(x−a) is done. This nice result is known as the remainder theorem.

Let's try it with our example. With $P\left(x\right)=x^2-5x+6$P(x)=x2−5x+6 and $D\left(x\right)=x-1$D(x)=x−1, before dividing note that $P\left(1\right)=\left(1\right)^2-5\left(1\right)+6=2$P(1)=(1)2−5(1)+6=2 and this is indeed the remainder!

 

The Factor Theorem

The factor theorem is an extension of the remainder theorem.

If a polynomial equation $P(x)=0$P(x)=0 has a root $x=a$x=a, meaning $P(a)=0$P(a)=0, then $x-a$x−a must be a factor of $P(x)$P(x). We could write $P(x)=(x-a)Q(x)$P(x)=(x−a)Q(x) where $Q$Q is a polynomial of degree one less than the degree of $P$P.

This means that if we can find by any means a number $a$a such that $P(a)=0$P(a)=0, then we know immediately that $x-a$x−a is a factor of $P$P.

Example

Factor $p(x)=x^3-x^2-x-2$p(x)=x3−x2−x−2.

By experiment, we find that $p(2)=0$p(2)=0. Therefore, $p(x)=(x-2)q(x)$p(x)=(x−2)q(x). We can use the division algorithm to discover $q(x)$q(x). That is, we divide $p(x)$p(x) by $x-2$x−2. In this way, we find that $q(x)=x^2+x+1$q(x)=x2+x+1 which is an irreducible quadratic. So, $p(x)=x^3-x^2-x-2=(x-2)(x^2+x+1)$p(x)=x3−x2−x−2=(x−2)(x2+x+1) is the complete factoring.

 

 

Worked Examples

Question 1

Question 2

Question 3