Polynomial expansion

Expansion of a polynomial expression can be obtained by repeatedly replacing subexpressions that multiply two other subexpressions, at least one of which is an addition, by the equivalent sum of products, continuing until the expression becomes a sum of (repeated) products.

Instead of multiplications, the expansion steps could also involve replacing powers of a sum of terms by the equivalent expression obtained from the binomial formula; this is a shortened form of what would happen if the power were treated as a repeated multiplication, and expanded repeatedly.

It is customary to reintroduce powers in the final result when terms involve products of identical symbols.

A more general single-step expansion will introduce all products of a term of one of the sums being multiplied with a term of the other: An expansion which involves multiple nested rewrite steps is that of working out a Horner scheme to the (expanded) polynomial it defines, for instance The opposite process of trying to write an expanded polynomial as a product is called polynomial factorization.

If both factors are binomials, the FOIL rule can be used, which stands for "First Outer Inner Last," referring to the terms that are multiplied together.