Simplifying Algebraic Fractions

Grade 6

The same idea as numeric fractions

Simplifying an algebraic fraction works exactly like simplifying a number fraction: find what is common on the top and bottom, then cancel it. The difference is that here you need to factorise first to spot what cancels.

You cannot cancel individual terms — you can only cancel factors. This is the trap most students fall into.

Worked example

EXAMPLE

Simplify: (x² + 5x + 6) / (x² + 3x + 2)

Factorise the top: (x + 2)(x + 3). Factorise the bottom: (x + 1)(x + 2). The factor (x + 2) appears on both top and bottom — cancel it. Answer: (x + 3) / (x + 1).

The cancelling trap

You cannot cancel like this: (x + 5) / (x + 3) ≠ 5/3. The x terms are not separate factors — they are part of a sum. Only cancel when the same bracket appears as a factor on top and bottom.

Multiplying and dividing algebraic fractions

To multiply algebraic fractions: multiply tops together, multiply bottoms together, then simplify. To divide: flip the second fraction and multiply. Factorise before you multiply — that way common factors are easy to spot and cancel early.

Try this one

Work out the value of \( \dfrac{y}{2} + \dfrac{y}{5} \) when \( y = 3.5 \).

[3]

Nova's hint:

This is substituting a value into a sum of two algebraic fractions.

Ask yourself: 'is it easier to work out each fraction separately and then add?' For instance, for \( \dfrac{t}{2}+\dfrac{t}{4} \) you could evaluate each term first, then combine.

Try to Solve it with Nova?