Thread: Polynomial questions

Hello. Have a few questions about polynomials that were never really cleared up in my algebra courses.

1. Are a polynomial's derivatives and antiderivatives (of all orders) also polynomials?
2. If a function is equal to a polynomial, but isn't in the ordinary form of one (as in, it looks nothing like the usual sum of powers with coefficients), is it still considered a polynomial function?
3. The proof of the rational root theorem assumes that and are coprime. Does this need to be taken into account when evaluating the roots this way? It never seems I've had to worry about it when simply using it, so why is it necessary in the elementary proof?

1. yes (trivially) integrals and derivatives of powers of x are also powers of x.

2. how do make a function which is equal to a polynomial for all values of the argument and avoid looking like a polynomial?


for 3. Rational root theorem - Wikipedia, the free encyclopedia

Originally Posted by mathman

2. How do make a function which is equal to a polynomial for all values of the argument and avoid looking like a polynomial?

For example, which is true for all reals. I was able to find this info on Wiki so this question's resolved.

for 3. Rational root theorem - Wikipedia, the free encyclopedia

Thanks, but I don't see anything that answers my question.

Those polynomials are called chebyshev polynomials epidecus - are are quite important in applied mathematics

Thanks for the insight, river rat

Luckily, I found the answer to #3. When I was browsing through the proof, I never put much thought to the fact that the assumption of the coprime operands was fundamental to the proof's method.

Originally Posted by epidecus

For example, which is true for all reals.

(Italics mine.)

The equality would seem to hold only for magnitudes of x less than or equal to 1.

Originally Posted by tk421

The equality would seem to hold only for magnitudes of x less than or equal to 1.

It holds for all reals. is complex valued for magnitudes of greater than 1. And the cosine function (apparently) seems to have real outputs for complex values.

Originally Posted by epidecus

Originally Posted by tk421

The equality would seem to hold only for magnitudes of x less than or equal to 1.

It holds for all reals. is complex valued for magnitudes of greater than 1. And the cosine function (apparently) seems to have real outputs for complex values.

Ah yes, of course. Thanks for reminding me of the relationships between the hyperbolic trig functions with real arguments and ordinary trig functions with imaginary arguments, and vice-versa.