Mr. Integral
Math notes/Pre-Calculus/Factoring

The Sum and Difference of Cubes — Cheat sheet

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Key method

A3−B3=(A−B)(A2+AB+B2).A^3-B^3=(A-B)(A^2+AB+B^2). A3+B3=(A+B)(A2−AB+B2).A^3+B^3=(A+B)(A^2-AB+B^2).

The sign in the first factor matches the original; the middle sign in the quadratic is opposite.

Example

Since 8x3+27=(2x)3+338x^3+27=(2x)^3+3^3, its factors are

(2x+3)(4x2−6x+9).(2x+3)(4x^2-6x+9).

Multiplying gives 8x3−12x2+18x+12x2−18x+278x^3-12x^2+18x+12x^2-18x+27: the middle terms cancel. This is why changing the quadratic's middle sign breaks the identity.

Avoid this mistake

A sum of cubes is not a cube of a sum: (A+B)3(A+B)^3 also contains two mixed terms.