# Factoring by decomposition: why it works

If a trinomial can be factored into two binomial terms, it can be factored using the method of decomposition, also known as ‘splitting the middle term’. Here is a trinomial that can be factored into two binomial terms:

We can confirm the factors by multiplying out the brackets.

The method of decomposition states that given a factorable trinomial of the form we can calculate the values in the brackets by finding two numbers that sum to the coefficient and that multiply to the product . On this page we explore why this method works.

## Compare coefficients

We begin with the assumption that our trinomial does indeed factor. Let’s generalise:

Since all trinomials can factor using irrational or complex numbers, we’re interested here in the case where and are integers.

If it is true that are integers then what I will show here is that there exist two related integers and such that and .

Let’s multiply out the right hand side:

Let’s compare the coefficients of the line

with

We notice that

Therefore for any trinomial that factors to the form , the coefficient is equal to .

Letting and , we calculate that .

We can conclude that any trinomial that factors to the form will have corresponding integers such that when .

We now have two conditions on the two unknowns and which guarantees that we will be able to find a unique value for and for . (Interestingly, solving this system simultaneously rather than by inspection results in a quadratic equation).

## Why does the grouping always work out?

In our example above, we have

According to our argument above, we are require to find two numbers and such that and

Examining the factors of 30 we find and or vice versa.

After splitting the middle term using we have

No matter which way around we set the and the , grouping the first two terms and the last two terms will yield a common factor. Let’s see:

or, vice versa

Either way, we see a common factor emerge.

Let’s examine the trinomial in terms of again. Given that

(this line comes from the assumption that the trinomial can be factored).

The right hand side can be arranged in two ways:

or

In both cases a common factor emerges:

and

Either way a common factor emerges and so we can complete the factoring process.

## The simple case

Notice that in the simple case, . That is the line

becomes

We still require two numbers such that

and

However, since in this case , the last line becomes

Or, as we are used to saying ‘two numbers that add to the coefficient b and multiply to the coefficient c’.