Combinatorial Methods in n- nomial Algebra and Their Application

The effectiveness of the application of the algebra of n-nomials to problems, whose solution causes a certain difficulty when using the formulas of ordinary (binomial) algebra, is shown. This fully confirms the prophetic words of the father of the theory of invariants D. Sylvester (John James Sylvester): "The part in some sense greater than the whole: general proposition must be proved easier than any partial case". The original concise combinatorial derivation of the formulas of abbreviated multiplication of $n-$nomials – $n-$term expressions $a_1+\cdots+a_n$ is given, which allows to translate many problems, that previously were difficult to solve by known methods, into the category of ordinary problems by means of the algebra of $n-$nomials. A special role to simplify the application of the formula for the sum of n cubes is that in the particular case of 3 variables when $a+b+c=0$, it is simplified to the form $a^3+b^3+c^3=3abc$, and this is the main point to eliminate the complexity of solving difficult problems. The most interesting thing is that this conditional identity admits a generalization to $n$ terms. So, if $a_1+\cdots+a_n=0$, then $a^3_1+\cdots+a^3_n=3\sum\limits_{1\le i<j<k\le n}a_{i}a_{j}a_{k}.$ The simplification effect is clearly demonstrated in many examples, where instead of the traditional cube formula ${(a_1\ +\ \cdots +\ a_n)}^3$ the following formula is used: the sum of cubes is tripled sum of $a_ia_ja_k(1\le i<j<k\le n)$, this is especially convenient when solving equations with cubic irrationalities and proofs of cubic ratios.