Exploring Algebraic Identities

Algebraic identities are true for every value of the letters. They speed up squaring, cubing and factorising.

Key ideas

• An identity is true for all values of its variables; an equation is true only for some.
• Expanding uses the identity left to right; factorising uses it right to left.
• Factorise fully: take out any common factor first, e.g. 16x² − 100 = 4(4x² − 25) = 4(2x + 5)(2x − 5).
• Identities make mental arithmetic easy: 103² = (100 + 3)² = 10609 and 98 × 102 = 100² − 2² = 9996.
• If a + b + c = 0, then a³ + b³ + c³ = 3abc.

Remember

Watch out

• Writing (a + b)² = a² + b²: the middle term 2ab is missing.
• Stopping before factorising fully (leaving a common factor inside a bracket).
• Sign slips in a³ − b³: the second bracket is a² + ab + b², all plus.

Worked example