Trinomial expansion
Encyclopedia
In mathematics
, a trinomial expansion is the expansion of a power of a sum of three terms into monomial
s. The expansion is given by
where n is a nonnegative integer and the sum is taken over all combinations of nonnegative indices i, j, and k such that i + j + k = n. The trinomial coefficients are given by
This formula is a special case of the multinomial formula for m = 3. The coefficients can be defined with a generalization of Pascal's triangle
to three dimensions, called Pascal's pyramid
or Pascal's tetrahedron.
The number of terms of an expanded trinomial is
where n is the exponent to which the trinomial is raised.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a trinomial expansion is the expansion of a power of a sum of three terms into monomial
Monomial
In mathematics, in the context of polynomials, the word monomial can have one of two different meanings:*The first is a product of powers of variables, or formally any value obtained by finitely many multiplications of a variable. If only a single variable x is considered, this means that any...
s. The expansion is given by
where n is a nonnegative integer and the sum is taken over all combinations of nonnegative indices i, j, and k such that i + j + k = n. The trinomial coefficients are given by
This formula is a special case of the multinomial formula for m = 3. The coefficients can be defined with a generalization of Pascal's triangle
Pascal's triangle
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients in a triangle. It is named after the French mathematician, Blaise Pascal...
to three dimensions, called Pascal's pyramid
Pascal's pyramid
In mathematics, Pascal's Pyramid is a three-dimensional arrangement of the trinomial numbers, which are the coefficients of the trinomial expansion and the trinomial distribution. Pascal's Pyramid is the three-dimensional analog of the two-dimensional Pascal's triangle, which contains the binomial...
or Pascal's tetrahedron.
The number of terms of an expanded trinomial is
where n is the exponent to which the trinomial is raised.
See also
- Binomial expansion
- Pascal's pyramidPascal's pyramidIn mathematics, Pascal's Pyramid is a three-dimensional arrangement of the trinomial numbers, which are the coefficients of the trinomial expansion and the trinomial distribution. Pascal's Pyramid is the three-dimensional analog of the two-dimensional Pascal's triangle, which contains the binomial...
- Multinomial coefficient