Algebraic fractions are one of those topics that can feel intimidating but quickly become manageable once you see the pattern. They appear on higher tier papers across all major boards — AQA (8300), Edexcel (1MA1), and OCR (J560) — typically in the later questions on Paper 2 or Paper 3, where they're worth 3–5 marks. Get confident with this topic and you'll be picking up marks that many students throw away.
What You Need to Know First
Before diving in, make sure you're solid on these:
- Factorising quadratics — you'll need this to simplify algebraic fractions (e.g. x² + 5x + 6 = (x + 2)(x + 3))
- Difference of two squares — recognise a² − b² = (a + b)(a − b) quickly
- Finding the lowest common multiple (LCM) — essential for adding or subtracting fractions
- Expanding brackets and collecting like terms — used throughout solving steps
- Basic fraction arithmetic — the logic is identical to numerical fractions; algebra just replaces the numbers
If any of these feel shaky, spend ten minutes reviewing them first. The algebraic fraction methods below will click much faster.
Simplifying Algebraic Fractions
Simplifying an algebraic fraction means cancelling common factors from the numerator and denominator — exactly like reducing 12/18 to 2/3 by cancelling the factor of 6. The key word is factors, not terms. You can only cancel something that multiplies the entire top or bottom.
Step-by-step method:
- Fully factorise the numerator.
- Fully factorise the denominator.
- Cancel any factors that appear in both.
- Write the simplified result.
Worked Example 1:
Simplify:
$$\frac{x^2 + 5x + 6}{x^2 + 3x + 2}$$
Step 1: Factorise the numerator. x² + 5x + 6 = (x + 2)(x + 3)
Step 2: Factorise the denominator. x² + 3x + 2 = (x + 1)(x + 2)
Step 3: Cancel the common factor (x + 2).
$$\frac{(x + 2)(x + 3)}{(x + 1)(x + 2)} = \frac{x + 3}{x + 1}$$
Answer: (x + 3) / (x + 1)
Worked Example 2:
Simplify:
$$\frac{x^2 - 9}{x^2 - x - 6}$$
Step 1: Factorise the numerator using the difference of two squares. x² − 9 = (x + 3)(x − 3)
Step 2: Factorise the denominator. x² − x − 6 = (x − 3)(x + 2)
Step 3: Cancel the common factor (x − 3).
$$\frac{(x + 3)(x - 3)}{(x - 3)(x + 2)} = \frac{x + 3}{x + 2}$$
Answer: (x + 3) / (x + 2)
Notice that in both examples, the entire first step is factorising. Students who go straight to cancelling without factorising first almost always make errors.
Adding, Subtracting and Solving with Algebraic Fractions
Adding and Subtracting
To add or subtract algebraic fractions, you need a common denominator — just as with numerical fractions. The safest common denominator is the product of both denominators (you can simplify afterwards if needed).
Step-by-step method:
- Identify (or construct) the lowest common denominator.
- Rewrite each fraction with that denominator, adjusting the numerator accordingly.
- Combine the numerators over the single denominator.
- Expand and simplify the numerator.
Worked Example 3:
Write as a single fraction: 3/(x + 2) + 5/(x − 1)
Step 1: Common denominator = (x + 2)(x − 1)
Step 2: Rewrite each fraction:
$$\frac{3(x - 1)}{(x + 2)(x - 1)} + \frac{5(x + 2)}{(x + 2)(x - 1)}$$
Step 3: Combine the numerators:
$$\frac{3(x - 1) + 5(x + 2)}{(x + 2)(x - 1)}$$
Step 4: Expand the numerator: 3x − 3 + 5x + 10 = 8x + 7
$$\frac{8x + 7}{(x + 2)(x - 1)}$$
Answer: (8x + 7) / [(x + 2)(x − 1)]
Solving Equations with Algebraic Fractions
When an equation contains algebraic fractions, the most efficient method is to multiply through by the common denominator to clear all fractions, then solve the resulting equation.
Worked Example 4:
Solve: 4/(x + 1) − 1/x = 1
Step 1: Common denominator = x(x + 1).
Step 2: Multiply every term by x(x + 1):
$$4x - (x + 1) = x(x + 1)$$
Step 3: Expand both sides: 4x − x − 1 = x² + x 3x − 1 = x² + x
Step 4: Rearrange to a quadratic equal to zero: 0 = x² + x − 3x + 1 0 = x² − 2x + 1 0 = (x − 1)²
Step 5: Solve: x = 1
Always check your answer back in the original equation. Here: 4/(1 + 1) − 1/1 = 2 − 1 = 1 ✓
Common Mistakes Students Make
- Cancelling terms instead of factors. Writing (x + 3)/(x + 5) and cancelling the x's to get 3/5 is wrong — x is not a factor of the numerator or denominator here.
- Forgetting to factorise before cancelling. Students often spot that both numerator and denominator contain x² and try to cancel that directly without factorising first.
- Sign errors when expanding brackets with a minus sign. In example 3, multiplying −1/x by x(x + 1) gives −(x + 1), not −x + 1. Watch the bracket.
- Not checking for values that make the denominator zero. If x = −2 makes the denominator zero, it must be excluded. Examiners won't always ask for this explicitly, but state it if the question asks for a full solution.
- Leaving the denominator unexpanded when it isn't needed. In adding/subtracting questions, leave the denominator in factored form unless the mark scheme asks you to expand — it's cleaner and less likely to introduce errors.
Exam Tips for Algebraic Fractions
- Always factorise both top and bottom before anything else. Even if you think you can spot the cancellation — write the full factorisation. Mark schemes award method marks for this step even if you then make an error.
- AQA often sets a "show that" question where you simplify and then use the result in the next part. A clean factorisation in part (a) makes part (b) much easier.
- On Edexcel 1MA1 higher papers, algebraic fractions are commonly combined with solving quadratics — the equation after multiplying through almost always factorises or requires the quadratic formula.
- Write every line of working. These questions are worth 4–5 marks, and the marks are awarded step by step. A wrong final answer with correct method shown can still earn 3 out of 4 marks.
- If your answer looks messy, go back and check the factorisation step — a messy result almost always means a factorisation error earlier on.
Practise Now
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