Binomial Theorem Nomenclature
Name: Bill G.
I am writing a paper about the binomial theorem and I am troubled by certain nomenclature. For
example, the expression (a + b) to the power of n is referred to as a binomial. Yet, when
expanded, it produces a polynomial.
It would be convenient if I could use the generic term binomial throughout my paper (as is done in
many textbooks) to refer to the compact form or the expanded version.
But, I do not feel comfortable doing this. Can you provide me with a justification for doing so?
For example, is there a general rule stating that any expression can always be referred to based on
the way it appears when written in the most compact form?
The formal definition of a polynomial, Pn(X), is: Pn(X) = An*X^n + An-1*X^n-1 + ... + A1*X + Ao.
Some of the coefficients may equal zero. The polynomial may also be compound, that is:
Pn,m(X,Y) = [An*X^n]*[Bm*Y^m] + ... + Ao*Bo. That X and/or Y can be
expressed as a polynomial itself X = (a+b) in your case is not at issue. So polynomial refers to
the highest value of the exponents 'n' and 'm' that appears in Pn,m(X,Y).
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Update: June 2012