Combinatorial Methods in n− nomial Algebra and Their Application

Abstract

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 a1 + · · · + an 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 a1 +· · ·+an = 0 , then a
3
1 +· · ·+a
3
n = 3 P
1≤i<j<k≤n
aiajak. The simplification
effect is clearly demonstrated in many examples, where instead of the traditional cube formula (a1 + · · · + an)
3
the
following formula is used: the sum of cubes is tripled sum of aiajak(1 ≤ i < j < k ≤ n) , this is especially convenient
when solving equations with cubic irrationalities and proofs of cubic ratios.