Trinomial Expansion
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High Quality Content by WIKIPEDIA articles! In mathematics, a trinomial expansion is the expansion of a power of a sum of three terms into monomials. When studying the structure of polynomials however, one often definitely needs a notion with the first meaning.With either definition, the set of monomials is a subset of all polynomials that is closed under multiplication. Both uses of this notion can be found, and in many cases the distinction is simply ignored, see for instance examples for the first and second meaning, and an unclear definition. In informal discussions the distinction is seld...
High Quality Content by WIKIPEDIA articles! In mathematics, a trinomial expansion is the expansion of a power of a sum of three terms into monomials. When studying the structure of polynomials however, one often definitely needs a notion with the first meaning.With either definition, the set of monomials is a subset of all polynomials that is closed under multiplication. Both uses of this notion can be found, and in many cases the distinction is simply ignored, see for instance examples for the first and second meaning, and an unclear definition. In informal discussions the distinction is seldom important, and tendency is towards the broader second meaning. When studying the structure of polynomials however, one often definitely needs a notion with the first meaning. This is for instance the case when considering a monomial basis of a polynomial ring, or a monomial ordering of that basis. An argument in favor of the first meaning is also that no obvious other notion is available to designate these values (the term power product is in use, but it does not make the absence of constants clear either), while the notion term of a polynomial unambiguously coindices with the second meaning of monomial. For an isolated polynomial consisting of a single term, one could if necessary use the uncontracted form mononomial, analogous to binomial and trinomial. The remainder of this article assumes the first meaning of "monomial".This formula is a special case of the multinomial formula for m = 3. It is of some interest that the coefficients are given by a generalization of Pascal's triangle to three dimensions, called Pascal's pyramid or Pascal's tetrahedron.
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