# Factoring a trinomial using decomposition

Factoring a trinomial such as can be tricky. There are several techniques, including inspection, using a grid, and decomposition – also known as ‘splitting the middle’. This page gives some examples of the decomposition technique.

## Review when :

Let’s expand :

We notice that gives us the ; and that gives us the constant .

To factor a trinomial of the form , we need two values such that and .

## Factor , where

To factor a trinomial where we need and .

First, let’s examine the expansion of a factored trinomial of this form:

This time, notice that as expected, but . Rather, .

Instead of looking for two numbers that add to 17 and multiply to 10, we need to look for two numbers that add to 17 and multiply to 30.

## Example 1:

Factor:

Look for two numbers, and , such that and .
The factor pairs of 40 are: (1,40); (2,20); (4,10); (5, 8)

To make 22, we need to use the pair (2,20).

We write a new line of work:

(It doesn’t matter if you write the first or the first).

Now we look for a common factor in the first two terms:

Perhaps suprisingly, (explained here), let’s look at the last two terms;

Ha! is a common factor! Put together we have:

Taking out as a common factor, we have:

## Example 2:

Find two numbers, and such that and .

Factor pairs of 24 are (1,24); (2,12); (3,8); (4,6). Now so let and .

Splitting the middle term with and we have:

Grouping we have:

Completing we have

## Example 3:

First, we notice that all three terms share a common factor 3. Let’s factor this out:

Now we factor the reduced trinomial ; however the is kept present throughout. You might also try factoring without reducing to see how the result compares.

Now we look for and such that , and .

Factor pairs of 8 are (1, 8); (2, 4). To satisfy , we need to use and .

Splitting the middle term we have

Grouping gives

Completing we have

## Try it out

With help – use the first applet below. No help? Use the second applet.

Correct fields show as green, incorrect as red.

Why method works

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