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Name: Bill G.
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Question:
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?



Replies:
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).

Vince Calder



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