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. 

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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. 

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

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. 

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

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

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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: 

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

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

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

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).

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

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