Expand: multiply every term
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 is the reverse — writing the expression back as brackets.
Take out the common factor first
Before anything else, factor out the largest number and the lowest power of each letter common to every term, e.g. .
Difference of two squares
. Spot it whenever one square is subtracted from another, e.g. .
Drawn from real examiner reports.
Signs swapped when factorising
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.
Reported on the November 2024 Higher Paper 1H (Q8a).
Sign slips when expanding
Mis-multiplying a term (e.g. writing instead of ) or failing to combine signed like terms such as .
Reported on the November 2024 Higher Paper 1H (Q3b) and June 2024 Higher Paper 1H (Q6a).
Not factorising fully
With the difference of two squares, forgetting a numerical factor — writing (losing the factor of 4) instead of .
Reported on the November 2024 Higher Paper 2H (Q23).
Only multiplying the first term
In a single bracket every inside term must be multiplied by the outside factor: , not . The must also be multiplied by .
(general exam technique)
Dropping the middle term of a square
Expanding as omits the middle term. — the four products give a term.
(general exam technique)
Taking out only part of the factor
Removing but not the from leaves , still factorisable. Take out the full common factor so nothing common remains inside.
(general exam technique)
Check by expanding back
Multiply your brackets out again as a quick self-check — it instantly catches a sign in the wrong place.
Do exactly what is asked
Expand removes brackets; factorise puts them back; simplify also collects like terms. A factorised answer to an expand question scores nothing — match the command word.
Collect like terms at the end
After a double bracket you have four terms; combine the two -terms so the answer is fully simplified, e.g. .
First thing to do when told to "factorise fully"?
Factorise fully .
