Revision notes, worked examples and methods for algebraic fractions.
Notes and quizzes ready · 50 questions · Revision video ready.
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.
Watch M18 · Algebraic fractions
Revise algebraic fractions with this narrated video. Use the player controls to pause, seek, adjust the volume or mute. Turn English captions on or off using the captions menu.
Mr Kidman's Classroom
