Algebraic Fractions

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 3𝑝 + 9 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. 

3𝑝 + 9 9𝑝 + 15 = 3(𝑝 + 3) 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. 

3(𝑝 + 3) 3(3𝑝 + 5) = 𝑝 + 3 3𝑝 + 5

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

Example 2
Simplify 3π‘₯Β² + 12π‘₯ π‘₯Β² – π‘₯ – 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. 

3π‘₯Β² + 12π‘₯ = 3π‘₯(π‘₯ + 4)
π‘₯Β² – π‘₯ – 20 = (π‘₯ – 5)(π‘₯ + 4)

Now, we can put this back into our fraction: 

3π‘₯Β² + 12π‘₯ π‘₯Β² – π‘₯ – 20 = 3π‘₯(π‘₯ + 4) (π‘₯ – 5)(π‘₯ + 4)

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

3π‘₯(π‘₯ + 4) (π‘₯ – 5)(π‘₯ + 4) = 3π‘₯ π‘₯ – 5

Example 3
Simplify 5𝑦² – 27𝑦 – 56 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).

5𝑦² – 27𝑦 – 56 𝑦² – 49 = (5𝑦 + 8)(𝑦 – 7) (𝑦 + 7)(𝑦 – 7)
5𝑦² – 27𝑦 – 56 𝑦² – 49 = 5𝑦 + 8 𝑦 + 7


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