The "Order" of Some Polynomials...

Factual Questions
1

When we speak of the “order” of a polynomial, can it ever be less than 0? For example, if y= x^(-2), what order would this equation be?

Also, consider this equation:
y = a+(bT)+(cT^3)+(d/ln(T)+(e*ln(T)/T)

where a,b,c,d,e = unique coefficients; T = temperature

Does this equation have an order? If so, would you call it a third order equation, I WAG? Is there a special name for such equations? (I ask because they show up a lot in engineering, for example.) Do such equations fall in the general category of a “series” of some kind? And, if so can a series be finite?

Thanks for your thoughts,

  • Jinx
2

The five-term equation you give there is definitely not a proper polynomial, so it does not have a polynomial order. A polynomial is formed by a linear combination of powers of the independent variable(s). Similarly, I don’t think that y = x[sup]-2[/sup] is a polynomial either, as it is commonly defined. However, I don’t see anything inconsistent about including negative powers or fractional powers in an alternate definition.

The equation involving ln(T) is that of a transcendental function. Transcendental functions can not be written in terms of finite polynomial series on T. So, if you were going to put it in terms of a polynomial, it would necessarily have infinite polynomial order.

Now, “order” is one of those words in mathematics that gets overused. For instance, differential equations are said to have an order that is different from polynomial order. It’s possible that that equation has an “order” in some other usage of the word.

3

Is a polynomial defined as y=ax^n+bx^(n-1)+cx^(n-2)…? Is that a proper polynomial? Hmm, I guess I was sick that day… I thought a polynomial was any function with multiple terms. Maybe I knew this once, but I have forgotten…

Thanks
Jinx

4

Strictly speaking, a polynomial is a sequence of coefficients such that every coefficient is zero after some point. As we all know, sequences are indexed by the natural numbers (including zero). So x[sup]-2[/sup] is right out.

You can do polynomial arithmetic without ever using variables, but it gets ugly. What’s (0, 1, 2, 1, …) * (1, 3, …)? Well, it’s (0, 1, 5, 7, 3, …), but I had to use x’s to do that. Check any introductory abstract algebra text for a full exposition.

5

y= x[sup]-2[/sup] is equivalent to x[sup]2[/sup]y - 1. Which is a bivariate polynomial.

6

A better definition than this one, and one which is probably easier to grasp than ultrafilter’s, is that a polynomial looks like this:

a[sub]n[/sub] x[sup]n[/sup] + a[sub]n-1[/sub] x[sup]n-1[/sup] + a[sub]n-2[/sub] x[sup]n-2[/sup] + … + a[sub]2[/sub] x[sup]2[/sup] + a[sub]1[/sub] x + a[sub]0[/sub]

A polynomial can be thought of as a finite sum of monomials, where a monomial looks like this:

a x[sup]n[/sup]

7

Except the first one is an equation, and the second one isn’t. There’s no way those two things are equivalent; they have different types. y = x[sup]-2[/sup] is equivalent to x[sup]2[/sup]y - 1 = 0, but that’s a bivariate polynomial equation, not a bivariate polynomial. And the fact that they’re equivalent doesn’t make x[sup]-2[/sup] a polynomial. It’s what we call a rational function, which is a function of the form p[sub]1/sub/p[sub]2/sub, where p[sub]1/sub and p[sub]2/sub are both polynomials.

8

Yeah, I agree with ultrafilter, but I can see where the confusion came in. The OP was not talking about:

the polynomial y = x[sup]-2[/sup]

but rather:

the polynomial y, which is equal to x[sup]-2[/sup].