Revision notes, worked examples and methods for algebraic fractions.
Notes and quizzes ready · 50 questions · Video coming soon.
This topic is Higher-only.
Revise the key ideas
Higher — simplifying algebraic fractions
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.
Higher — the four operations
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.
Higher — solving equations
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.
Test yourself
50 questions · Sets of 10 from the selected tier. For fractions, use / when typing; for powers, use superscripts or ^. Follow each question's answer format. These quick checks support revision; practise full written solutions and proofs too.
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