Simplifying Algebraic Fractions

Example 1:

Simplify:

a)

Step 1: Factorise the numerator.

The expression can be factorised by taking out the common factor since both terms contain

So,

Step 2: Cancel any common factors that appear in the numerator and denominator.

The factor appears in both the numerator and denominator, so it will cancel.

Note: Students get carried away with cancelling terms before factorising. has two terms on the numerator, and so we can’t cancel yet since we need one term on the numerator and denominator (this would be breaking our laws of BIDMAS).

Factorising turns the numerator into a product of factors which then becomes one term which can then cancel with other terms (for example, is two terms whereas is one term).

b)

Step 1: Factorise the denominator.

The expression is an example of the difference of two squares principle. Recall that

Using this principle, we factorise the denominator.

So,

Step 2: Cancel any common factors that appear in the numerator and denominator.

The factor appears in both the numerator and denominator, so it will cancel.

Example 2:

Simplify:

a)

Step 1: Factorise the denominator.

First, we look at what multiplies to give Since we write in both brackets.

Next, we look for two numbers that multiply to give and add to give We list the factor pairs of

The pairs are:

and

and

Out of these pairs, only and can make To get we do Therefore, the denominator factorises to

Therefore,

Step 2: Cancel any common factors that appear in the numerator and denominator.

The factor appears in both the numerator and the denominator, so it will cancel.

b)

Step 1: Factorise both the numerator and the denominator.

For the numerator:

First, we look at what multiplies to give Since we write in both brackets.

Next, we look for two numbers that multiply to give and add to give We list the factor pairs of

The pairs are:

and

and

and

and

Out of these pairs, only and can make To get we do Therefore, the denominator factorises to

We now apply the same process to factorise the denominator:

Like above, since we write in both brackets.

Next, we look for two numbers that multiply to give and add to give We list the factor pairs of

The pairs are:

and

and

Out of these pairs, only and can make To get we do Therefore, the denominator factorises to

Therefore,

Step 2: Cancel any common factors that appear in the numerator and denominator.

The factor appears in both the numerator and the denominator, so it will cancel.

Example 3:

Simplify:

Step 1: Factorise both the numerator and the denominator.

Here, we can start with the denominator because that is the easier quadratic expression to factorise.

We first notice that is a common factor in both terms, so we can factorise that out first.

Then, we use the difference of two squares principle to factorise the expression inside the brackets.

Now, we factorise the numerator.

First, we look at what multiplies to give This is and so we write these in the brackets.

Next, we look for two numbers that multiply to give We list the factor pairs of

and

and

Usually, we should first choose the pair whose values are closest together, which will be and We then decide which bracket the and should go in. The options are

or

We now apply the OI-OI method (see our Factorising Quadratics note for more details on this method) to check which arrangement gives the correct middle term, which is Let’s choose the first option.

Here, the outer product is and the inner product is We can make by doing Therefore, the numerator factorises to

Therefore,

Step 2: Cancel any common factors that appear in the numerator and denominator.

The factor appears in both the numerator and the denominator, so it will cancel.