Name the pattern you can see
The difference of two squares is often familiar; the difference of two cubes deserves the same fluency. Check an identity by multiplying its factors, then practise spotting it when the terms have been rearranged.
a³ − b³ = (a − b)(a² + ab + b²)
a³ + b³ = (a + b)(a² − ab + b²)
- Record excluded values before cancelling anything in a fraction.
- Factor the whole numerator and denominator; cancel common factors, not separate terms.
- Check by multiplying back or substituting a permitted value. A numerical check can catch an error, but is not a proof of an identity.
Worked example
A difference of cubes inside a fraction
Simplify (x³ − 8)/(x² − 4), stating all excluded values.
The original denominator vanishes at x = 2 and x = −2, so both are excluded. Factor:
x³ − 8 = (x − 2)(x² + 2x + 4)
x² − 4 = (x − 2)(x + 2).
For permitted values of x, cancel the common factor x − 2. The result is (x² + 2x + 4)/(x + 2), with x ≠ ±2.
The simplified formula happens to have a value at 2, but that does not put 2 back into the domain of the original expression.
Your turn
Use an identity without solving for x
A positive real number x satisfies x + 1/x = 3. Find x³ + 1/x³.
Show a hint
Expand (a + b)³ and group the middle two terms as 3ab(a + b).
Show the full solution
The identity is a³ + b³ = (a + b)³ − 3ab(a + b). Put a = x and b = 1/x. Then ab = 1 and a + b = 3.
x³ + 1/x³ = 3³ − 3 × 1 × 3 = 18.
Solving a quadratic for x would work, but it introduces square roots that cancel later. The symmetric expression lets you avoid that work.
Say which identity you used and why it avoided finding x. Then try the same argument with x + 1/x = k.
What to practise next
Build fluency with factorisation and the factor theorem, then practise expressions where the useful pattern is hidden.
- Physics & Maths Tutor: algebra and functions
Start with factor theorem, quadratics and algebraic-methods resources; use the matching solutions.
- TMUA papers and community practice
Apply the same habits to unfamiliar algebra questions in a mixed paper.
The examples and explanations on this page are part of Arij Asad’s teaching guides. Linked papers and worksheets belong to their named creators and publishers. PMT’s topic collections include exam-board questions and other credited materials.