Definition of a PolynomialDate: 12/04/97 at 17:06:23 From: Richard Subject: Definition of a Polynomial My daughter had a problem that said "are the following expressions polynomials?" Most were straightforward but one was just the number 5. She said no and I agreed. She got it wrong. I called a professor of math at a local university, who explained that it is a polynomial because it is 5 times x^0 (x to zero power.) Okay, I get that. But one of the other problems was just x. The answer in the back of the book says no, x alone is not a polynomial. How is just x different from just 5? Thanks! Date: 12/04/97 at 18:58:58 From: Doctor Pete Subject: Re: Definition of a Polynomial Hi, I'd say the book was wrong. For a more detailed answer, most mathematicians would define a polynomial this way: A polynomial of one variable is a function P[x] of the form a[0] + a[1]x^1 + a[2]x^2 + ..., where the coefficients a[0], a[1], ... may be 0. The largest value of n for which a[n] is not zero is called the degree of the polynomial. More generalized polynomials occur when we replace x with some function of x, say Sin[x] or Exp[x]. There are also polynomials of more than one variable, which you can guess have individual terms of the form a[k] x1^(p1) x2^(p2) ... xm^(pm). Clearly, the value x falls under this category (a[1] = 1, a[k] = 0 otherwise), and so does 5 (a[0] = 5, a[k] = 0 otherwise). So does 1+x+x^2+x^3+x^4+... = 1/(1-x), which we usually don't like to call a polynomial since it has infinite degree. Rather, we call it a power series. -Doctor Pete, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 12/05/97 at 01:05:40 From: Richard Subject: Re: Definition of a Polynomial Thanks. Much useful information. Really appreciate the quick response. Thanks again! |
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