M18 · Algebraic fractions | Edexcel GCSE Maths

Revise the key ideas

Factorise numerator and denominator before cancelling common factors. x² − 9x + 3 = (x − 3)(x + 3)x + 3 = x − 3, for x ≠ −3.
Cancel factors, not individual terms in sums. x + 3x is 1 + 3x, not 3. The denominator cannot be zero.
Record restrictions from the original denominator even when its factors disappear after cancellation. A simplified expression does not restore excluded values.
Worked example: 6x²9x = 2x3, with x ≠ 0. Divide coefficients by 3 and cancel a common factor x.

To multiply, factorise and cancel across the complete numerator and denominator: x² − 4x + 1 × x + 1x − 2 = x + 2, with x ≠ −1 or 2.
To divide, multiply by the reciprocal of the second fraction. Check that the divisor itself is not zero as well as checking all denominators.
Worked example: x3 ÷ x²6 = x3 × 6x² = 2x, where x ≠ 0.
For addition or subtraction, use a common denominator and combine whole numerators. 2x + 3x + 1 = 2(x + 1) + 3xx(x + 1) = 5x + 2x(x + 1).
Worked example: 1x − 1 − 1x + 1 = (x + 1) − (x − 1)(x − 1)(x + 1) = 2x² − 1, with x ≠ ±1. Bracket the subtracted numerator.

Multiply both sides of an equation by a common denominator to remove fractions, while preserving the restrictions on x.
Worked example: 2x = 3x + 1 gives 2(x + 1) = 3x, so x = 2. This is allowed since x is neither 0 nor −1.
Solutions obtained after clearing denominators must be checked in the original equation. Reject values that cause division by zero, even if they satisfy the rearranged polynomial.
If a quadratic appears, solve it by factorisation or the quadratic formula and test every candidate against the domain.

Practice questions with explained answers will be added here.

For now, cover the worked answers, try the calculations yourself, then compare each step. Include units and reasons where needed.