Expand = multiply out; factorise = reverse
To expand multiply by both terms. For two brackets multiply each term in the first by each in the second (four products), then collect like terms. Factorising puts an expression back into brackets.
Take out the highest common factor first
Before anything else, factor out the largest number and the lowest power of each letter common to every term, e.g. .
Recognise the difference of two squares
. Spot it whenever you see one square subtracted from another, e.g. .
Drawn from real examiner reports.
Not factorising fully
With the difference of two squares, students forget a numerical factor, e.g. writing (which loses the factor of 4) instead of . Pull out the common factor before spotting the difference of two squares.
A commonly reported examiner note records this on a higher-tier paper.
Slips when expanding with negatives
Mis-multiplying a term (e.g. writing instead of ), or failing to combine signed like terms such as . A negative outside a bracket flips every sign inside it.
A commonly reported examiner note records this across recent higher-tier papers.
Signs the wrong way round in factors
A correct factor pair is chosen but the signs are swapped, so the brackets expand back to the wrong expression — e.g. giving when was needed.
A commonly reported examiner note records this on a recent higher-tier paper.
Adding indices when adding terms
You only add indices when multiplying powers of the same base, never when adding terms. So — those are unlike terms and cannot be combined at all; the most you can do is factorise: .
Expanding three brackets in pairs
For , expanding two different pairs separately and adding them scores no marks. Expand one pair into a quadratic first, then multiply that quadratic by the remaining bracket and collect like terms.
Check factors by expanding them back out
Examiners recommend multiplying your brackets out again as a quick self-check — it instantly catches a sign in the wrong place.
Reuse your factors when it says 'hence'
"Factorise, hence solve" means use the factors you just found — set each factor equal to , e.g. gives or . Do not restart with the quadratic formula; that ignores the "hence".
Expand two brackets first, then the third
For a product of three brackets, expand any two into a quadratic, then multiply that quadratic by the third bracket. This keeps every term accounted for and avoids the zero-scoring "add the pairs" route.
Quick way to check your factors are right?
Factorise fully .
