# Algebraic Fractions

Just like with numerical fractions, simplifying algebraic fractions is simply about finding common factors in the numerator and denominator. The only difference is how we find the factors – with algebraic expressions, we have to factorise them first.

Example 1
Simplify $\frac{3𝑝 + 9}{\mathrm{9𝑝 + 15}}$ fully.

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We need to start by factorising both the numerator and denominator. In this case, we can take out a factor of 3 from both.

$\frac{3𝑝 + 9}{\mathrm{9𝑝 + 15}}=\frac{3\left(𝑝 + 3\right)}{\mathrm{3\left(3𝑝 + 5\right)}}$

It’s now easy to see that the numerator and denominator have a common factor of 3, so we can divide both the numerator and denominator by 3.

$\frac{3\left(𝑝 + 3\right)}{\mathrm{3\left(3𝑝 + 5\right)}}=\frac{𝑝 + 3}{\mathrm{3𝑝 + 5}}$

Since we took out the highest common factor at the beginning, the fraction is in its simplest form.

Example 2
Simplify $\frac{3𝑥² + 12𝑥}{\mathrm{𝑥² – 𝑥 – 20}}$ fully.

The process is exactly the same as in example 1; we just have slightly more complex expressions to factorise. It can help to consider them separately, rather than in the fraction.

3𝑥² + 12𝑥 = 3𝑥(𝑥 + 4)
𝑥² – 𝑥 – 20 = (𝑥 – 5)(𝑥 + 4)

Now, we can put this back into our fraction:

$\frac{3𝑥² + 12𝑥}{\mathrm{𝑥² – 𝑥 – 20}}=\frac{3𝑥\left(𝑥 + 4\right)}{\mathrm{\left(𝑥 – 5\right)\left(𝑥 + 4\right)}}$

As you can see, (𝑥 + 4) is common to the numerator and denominator so we can divide both by (𝑥 + 4).

$\frac{3𝑥\left(𝑥 + 4\right)}{\mathrm{\left(𝑥 – 5\right)\left(𝑥 + 4\right)}}=\frac{3𝑥}{\mathrm{𝑥 – 5}}$

Example 3
Simplify $\frac{5𝑦² – 27𝑦 – 56}{\mathrm{𝑦² – 49}}$ fully.

Just as before, we need to factorise the numerator and denominator. This time, it’s easier to factorise the denominator so we’ll start with that.

𝑦² – 49 = (𝑦 + 7)(𝑦 – 7)

Now, we can assume that (𝑦 + 7) or (𝑦 – 7) is a factor of the numerator – this makes it much easier to factorise.

5𝑦² – 27𝑦 – 56 = (5𝑦 + 8)(𝑦 – 7)

This time, (𝑦 – 7) is common to the numerator and denominator so we can divide both by (𝑦 – 7).

$\frac{5𝑦² – 27𝑦 – 56}{\mathrm{𝑦² – 49}}=\frac{\left(5𝑦 + 8\right)\left(𝑦 – 7\right)}{\mathrm{\left(𝑦 + 7\right)\left(𝑦 – 7\right)}}$
$\frac{5𝑦² – 27𝑦 – 56}{\mathrm{𝑦² – 49}}=\frac{5𝑦 + 8}{\mathrm{𝑦 + 7}}$

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