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The Higher-tier algebra skill that combines factorising with fraction operations. Complete guide with KFC method, LCD technique and worked examples for grades 7-9.
Algebraic fractions are fractions where the numerator, the denominator, or both contain algebraic expressions. At IGCSE Higher tier, you must be able to simplify them by factorising and cancelling, multiply and divide them using the KFC method, and add and subtract them by finding a lowest common denominator. These skills are essential for solving equations and working with functions.
Source: Edexcel IGCSE Mathematics (9-1) Specification 4MA1
Simplifying algebraic fractions follows the same logic as simplifying numerical fractions: divide the numerator and denominator by their common factors. The key difference is that you must factorise first.
Remember: you can only cancel factors (expressions that are multiplied), not individual terms (expressions that are added or subtracted). For example, in (x² + 5)/(x² + 2) you cannot cancel the x² terms because they are being added, not multiplied.
To multiply algebraic fractions, multiply the numerators together and the denominators together. Before multiplying, factorise everything and cross-cancel common factors between any numerator and any denominator. This keeps expressions manageable and avoids unnecessary expansion.
To divide algebraic fractions, use the KFC method:
Then proceed as with multiplication: factorise, cross-cancel, and multiply.
Adding and subtracting algebraic fractions requires a common denominator, just like with numerical fractions. Follow these four steps:
Important: Do NOT expand the denominator. Leave it in factored form. Expanding creates unnecessary work and increases the chance of error.
Simplify: (2x² − 5x − 3) ÷ (x² − 9)
Step 1 (Factorise numerator): 2x² − 5x − 3 = (2x + 1)(x − 3)
Step 2 (Factorise denominator): x² − 9 = (x − 3)(x + 3) [DOTS]
Step 3 (Cancel common factor): (x − 3) appears in both, so cancel it
Step 4 (Write result): = (2x + 1) ÷ (x + 3)
Answer: (2x + 1)/(x + 3)
Simplify: x/(3x − 6) ÷ x²/(x − 2)
Step 1 (KFC — Keep, Flip, Change): x/(3x − 6) × (x − 2)/x²
Step 2 (Factorise): 3x − 6 = 3(x − 2), so the expression becomes x/(3(x − 2)) × (x − 2)/x²
Step 3 (Cancel common factors): (x − 2) cancels, and x/x² leaves 1/x
Step 4 (Multiply): = 1/(3x)
Answer: 1/(3x)
Write as a single fraction: 4/(x + 2) − 3/(2x − 1)
Step 1 (Find LCD): LCD = (x + 2)(2x − 1)
Step 2 (Adjust numerators): 4(2x − 1)/((x + 2)(2x − 1)) − 3(x + 2)/((x + 2)(2x − 1))
Step 3 (Expand numerators): (8x − 4 − 3x − 6) / ((x + 2)(2x − 1))
Step 4 (Simplify numerator): (5x − 10) / ((x + 2)(2x − 1))
Step 5 (Factorise numerator): 5(x − 2) / ((x + 2)(2x − 1))
Answer: 5(x − 2)/((x + 2)(2x − 1))
Fake cancellation: You cannot cancel terms from (x² + 5)/(x² + 2) because the x² terms are added, not multiplied. Only factors can be cancelled. Always factorise first.
Sign errors when subtracting: In 4/(x + 2) − 3/(2x − 1), the minus applies to the entire second numerator: −3(x + 2) = −3x − 6, not −3x + 6. Use brackets around the second numerator to avoid this trap.
Expanding the denominator: Leave the denominator in factored form. Expanding it creates larger expressions and makes it harder to simplify at the end.
Forgetting to factorise before cancelling: Students try to cancel terms directly from the expression without factorising. This leads to incorrect simplification every time.
Always factorise before cancelling: This is the single most important rule. Factorise the numerator and denominator completely before looking for common factors to cancel.
Watch for hidden DOTS: Expressions like x² − 16 or 4x² − 25 are differences of two squares. Spotting these quickly makes factorising much faster.
Check by substituting a test value: Substitute a simple value (e.g. x = 2) into both the original expression and your simplified answer. If they give the same number, your simplification is correct.
Use brackets when subtracting: Write the second numerator in brackets with the minus sign outside. This prevents the most common sign error on this topic.
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