What is the formula for the sum of cubes?

The sum of cubes formula, ( a^3 + b^3 ), can be expressed as the product of a binomial and a trinomial: ( (a + b)(a^2 - ab + b^2) ). This relationship arises from polynomial identities and can be verified through algebraic expansion.

When you take the binomial ( (a + b) ) and multiply it by the trinomial ( (a^2 - ab + b^2) ), you utilize the distributive property, also known as the FOIL method (First, Outside, Inside, Last). This multiplication yields:

1. The first term: ( a(a^2) = a^3 )

2. The outer term: ( a(-ab) = -a^2b )

3. The inner term: ( b(a^2) = ab^2 )

4. The last term: ( b(-ab) = -ab^2 )

Combining these results, the terms ( -a^2b + ab^2 ) cancel each other when collected with ( -ab ). Thus you only keep ( a^3 + b^3 ) from the first term.