Three identities do most of the work in an Olympiad algebra section. Learn them in both directions — expanding and factorising — because the paper uses both.
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An identity is true for every value of the letters — unlike an equation, which is true only for some.
📜 Learn these cold
| Identity | Example |
|---|---|
| (a + b)² = a² + 2ab + b² | (x + 3)² = x² + 6x + 9 |
| (a − b)² = a² − 2ab + b² | (x − 5)² = x² − 10x + 25 |
| (a + b)(a − b) = a² − b² | (x + 7)(x − 7) = x² − 49 |
| (x + a)(x + b) = x² + (a+b)x + ab | (x + 2)(x + 5) = x² + 7x + 10 |
(a + b)² is not a² + b². The middle term 2ab is where nearly every dropped mark lives. Check with numbers: (2 + 3)² = 25, but 4 + 9 = 13.
Mental arithmetic with identities. 98 × 102 = (100 − 2)(100 + 2) = 10000 − 4 = 9996. And 102² = 10000 + 400 + 4 = 10404. Olympiad papers ask these as arithmetic and expect you to see the algebra.
Multiply every term of the first bracket by every term of the second, then collect like terms.
📤 The order of work
Multiply the numbers, then the letters: 3x × 4x² = 12x³.
Add the powers of the same letter — never multiply them.
Collect only like terms: 5x + 3x = 8x, but 5x + 3x² stays as it is.
Watch the signs: a minus in front of a bracket flips every term inside.
−(x − 4) = −x + 4, not −x − 4. The minus applies to everything in the bracket.
Running the identities backwards. Always take out the common factor first.
📥 The order to try
Common factor first. 6x² + 9x = 3x(2x + 3).
Difference of squares? Two terms with a minus between them: x² − 64 = (x + 8)(x − 8).
Perfect square? x² + 12x + 36 = (x + 6)².
Otherwise split the middle term. For x² + 7x + 12 find two numbers multiplying to 12 and adding to 7 — that is 3 and 4, giving (x + 3)(x + 4).
The favourite Olympiad twist. If a + b = 7 and ab = 12, then a² + b² = (a + b)² − 2ab = 49 − 24 = 25. Similarly if x + 1/x = 5 then x² + 1/x² = 25 − 2 = 23.
12 Olympiad questions on expanding and factorising. 🍀
