Simplify by factorising top and bottom, then cancel
Factorise the numerator and denominator fully (watch for the difference of two squares) and cancel only common factors — never individual terms.
Add/subtract over a common denominator
Use a common denominator, combine the numerators carefully (mind the signs when subtracting), then simplify.
Divide by multiplying by the reciprocal
To divide algebraic fractions, flip the second fraction and multiply; then cancel.
Drawn from real examiner reports.
Cancelling terms instead of factors
Cancelling an or a number that is only part of a sum (e.g. cancelling the in ) instead of cancelling a whole bracket.
Sign errors when subtracting a fraction
Forgetting that subtracting a fraction means subtracting all of its numerator, e.g. mishandling .
Not multiplying every term when solving
When clearing denominators to solve, students multiply some terms but not others (e.g. forgetting to multiply a term on the right-hand side).
Difference of two squares with a leading coefficient
Treating as drops the factor of — it is . Pull out the common factor first (or factor as and then take the out of each bracket).
Dropping the x (or a bracket) when combining over a common denominator
Two classic slips: writing the final answer over a constant denominator (e.g. dividing top and bottom by but only on the top), and not bracketing a subtracted numerator so the second sign is not flipped.
Factorise first — even on a hard question
On a tough algebraic-fractions question, factorising any quadratics (and spotting a difference of two squares) usually unlocks cancelling and earns method marks.
Factorise every quadratic FIRST, then cancel
On a "simplify"/"show that" algebraic-fraction finale, factorise each numerator and denominator before doing anything else. Cancelling common brackets collapses the expression; multiplying brackets out first produces an unmanageable quartic and almost never recovers.
Factorise top and bottom fully, then cancel common brackets: You may cancel a whole factor like — never cancel a term inside a sum (you cannot cancel the or the ).
Factorise 16x² − 36 fully.
Simplify .
