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Factoring formulas

Perfect cube with a plus sign

x³ + 3ax² + 3a²x + a³ = (x + a)³
x³ + 3ax² + 3a²x + a³=(x + a)³recognise the pattern, then replace it with the factor form

The coefficients 1, 3, 3, 1 come from expanding (x + a) three times.

Detailed notes

Formula Walkthrough

Cubic identities are patterns for expanding or factorising expressions involving cubes.

Step 1: Perfect cube with plus

Expanding (x + a) three times creates the coefficient pattern 1, 3, 3, 1.

(x + a)3 = x3 + 3ax2 + 3a2x + a3

Step 2: Perfect cube with minus

When the bracket is x - a, the signs alternate.

(x - a)3 = x3 - 3ax2 + 3a2x - a3

Step 3: Sum of cubes

A sum of cubes has a simple factor x + a and a quadratic factor with alternating signs.

x3 + a3 = (x + a)(x2 - ax + a2)

Step 4: Difference of cubes

A difference of cubes has x - a as the simple factor, and the quadratic factor uses all plus signs.

x3 - a3 = (x - a)(x2 + ax + a2)