Factorizing Polynomials

Factorizing Polynomials & The Remainder Theorem

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You should already be familiar with basic factorizing techniques,( common factors and quadratics) so these will not be covered here.

What we will look at is how to factorize higher expressions ( such as cubics ) and how to divide algebraically.

Suppose you were asked to factorize the expression

and you were told that one factor was

Factorizing in this case means dividing one expression by the other. In cases like this, where you know that the expressions divide exactly, it is not necessary to do a full long division. The result can easily be deduced. We now that after factorizing we will have an expression like

it is just a case of working out a, b and c.

We want

clearly a must equal 1, which gives

Now we must make sure that we get the right number of x² terms. So far we are going to get 2, so we need to pick up another 3. Therefore b must equal 3. We now have

Now consider the x terms. So far we have 6, which means we need to find another 2. c must therefore equal 2. So we have

The second bracket is just a quadratic and is easily factorized. The final result is therefore.

This method of factorizing is extremely easy, once you have worked out what is going on. There are therefore 2 more examples below.

Divide
by

The only way to get 3x⁴ is to start the bracket with 3x³

Looking at the x³ terms, we so far have -6, which means that we need to pick up +2. Therefore b must equal +2

Now looking at the x² terms, we currently have -4, so we need to make up +1. Therefore c must equal 1.

Considering the x terms, we currently have -2, but should not have any, so d must equal 2. This gives

You can check that you have not made a mistake, by seeing if you have the right number at the end. We do get -4, so our answer is correct.

Divide
by

In order to get 4x³ the bracket must start with 2x².

Looking at the x² terms, we currently have +2, but we need -4 and so must lose 6. Therefore b must equal -3.

Now looking at the x terms, we have -3 and must therefore get another -2. c must equal -1.

Again, check that this gives the correct number on the end (-1).

On the next page, we will look at long division and remainders, but for straightforward factorizing and division, I would recommend the method shown on this page.

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