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Class 9 Exploring Algebraic Identities: cheat sheet
CBSE Mathematics. 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
| Squares | (a + b)² = a² + 2ab + b²; (a − b)² = a² − 2ab + b² |
| Difference of squares | a² − b² = (a + b)(a − b) |
| Three terms | (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca |
| Cubes | (a + b)³ = a³ + b³ + 3ab(a + b); (a − b)³ = a³ − b³ − 3ab(a − b) |
| Sum and difference of cubes | a³ + b³ = (a + b)(a² − ab + b²); a³ − b³ = (a − b)(a² + ab + b²) |
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
Without cubing, find 10³ + 6³ + (−16)³.
Working: the numbers add to 0, so the sum of cubes is 3 × 10 × 6 × (−16)
Answer: −2880
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