Algebraic fractions are a Higher tier topic that catches out a lot of confident students because the arithmetic feels familiar but the cancelling rule is unforgiving — get it wrong by half a step and the whole question collapses. This guide walks through the four operations and the single most-tested skill in the topic: knowing what you can cancel and what you absolutely can't.
The cancelling rule — the only rule that matters
You can only cancel a factor, never a term. A factor is something you've multiplied; a term is something you've added.
(x² + 3x) / x looks unhelpful; written as x(x + 3) / x it's clearly x + 3. Factorising is what makes the cancelling visible.
Simplifying algebraic fractions
Step 1: factorise top and bottom. Step 2: cancel any factor that appears on both.
Worked example 1
Simplify (x² − 9) / (x² + 5x + 6).
Top: difference of two squares — (x − 3)(x + 3).
Bottom: factorise the quadratic — two numbers that multiply to 6 and add to 5: 2 and 3. So (x + 2)(x + 3).
[(x − 3)(x + 3)] / [(x + 2)(x + 3)]
Cancel the (x + 3) from top and bottom:
= (x − 3) / (x + 2)
Adding and subtracting algebraic fractions
Same rule as numerical fractions: you need a common denominator. Multiply each fraction's top and bottom by what the other denominator gives you.
Worked example 2
Express 2/(x + 1) + 3/(x − 2) as a single fraction.
Common denominator is (x + 1)(x − 2).
= [2(x − 2) + 3(x + 1)] / [(x + 1)(x − 2)]
= [2x − 4 + 3x + 3] / [(x + 1)(x − 2)]
= (5x − 1) / [(x + 1)(x − 2)]
Multiplying algebraic fractions
Multiply tops; multiply bottoms. Cancel any factor common to a top and a bottom before multiplying — it makes the algebra cleaner.
Worked example 3
Simplify (x² − 4) / (3x) × (6x²) / (x − 2).
Factorise: x² − 4 = (x − 2)(x + 2).
[(x − 2)(x + 2)] / 3x × 6x² / (x − 2)
Cancel (x − 2) and simplify 6x² / 3x = 2x:
= (x + 2) × 2x = 2x(x + 2)
Dividing algebraic fractions
Same as dividing numerical fractions — flip the second fraction and multiply.
Worked example 4
Simplify (x + 5) / (x − 1) ÷ (x² + 5x) / (x² − 1).
Flip the second fraction:
(x + 5)/(x − 1) × (x² − 1)/(x² + 5x)
Factorise: x² − 1 = (x + 1)(x − 1) and x² + 5x = x(x + 5).
= (x + 5)/(x − 1) × [(x + 1)(x − 1)] / [x(x + 5)]
Cancel (x + 5) and (x − 1):
= (x + 1) / x
The mistakes that cost the most marks
(x + 5)/x can NOT be simplified to 5 + 1. The x is a term in the numerator, not a factor. Same for (x² + 3) / x² — the answer is not 1 + 3.
2/x + 3/y is NOT 5/(x+y). You need a common denominator first.
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