Trinome

In mathematics, the trinomial is the three-part equivalent of the better-known binomial . So it is a polynomial that is a sum of three monomials . The term trinomial formula also exists as a counterpart to the binomial formulas and in the standard case is related to a power of the species . ${\displaystyle (a+b+c)^{n}}$

Examples of trinomials:

${\displaystyle a+b+c}$,

${\displaystyle a^{2}+b^{2}-c^{2}}$,

${\displaystyle 3a^{2}-7ab^{3}+2ab^{4}}$

The square of the trinomial is: ${\displaystyle a+b+c}$

${\displaystyle (a+b+c)^{2}=a^{2}+b^{2}+c^{2}+2ab+2ac+2bc}$

The formula for any exponent is: ${\displaystyle n\in \mathbb {N} }$

${\displaystyle {\begin{alignedat}{4}(a+b+c)^{n}&=\sum _{i,j,k\geq 0 \atop i+j+k=n}{n \choose i,j,k}\,a^{i}\,b^{j}\,c^{k}\\&=\sum _{i=0}^{n}\sum _{j=0}^{n-i}{\frac {n!}{i!\,j!\,(n-i-j)!}}\cdot a^{i}\,b^{j}\,c^{n-i-j}\end{alignedat}}}$

The coefficients of the general power are also called trinomial coefficients and can be found in Pascal's pyramid . ${\displaystyle (a+b+c)^{n}}$

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Wiktionary: Trinom  - explanations of meanings, word origins, synonyms, translations