Trinomial expansion
In mathematics, a trinomial expansion is the expansion of a power of a sum of three terms into monomials. The expansion is given by
where is a nonnegative integer and the sum is taken over all combinations of nonnegative indices and such that. The trinomial coefficients are given by
This formula is a special case of the multinomial formula for. The coefficients can be defined with a generalization of Pascal's triangle to three dimensions, called Pascal's pyramid or Pascal's tetrahedron.
Derivation
The trinomial expansion can be calculated by applying the binomial expansion twice, setting, which leads toAbove, the resulting in the second line is evaluated by the second application of the binomial expansion, introducing another summation over the index.
The product of the two binomial coefficients is simplified by shortening,
and comparing the index combinations here with the ones in the exponents, they can be relabelled to, which provides the expression given in the first paragraph.
Properties
The number of terms of an expanded trinomial is the triangular numberwhere is the exponent to which the trinomial is raised.