Polynomials do or don't have integer roots?

[povatix]

Homework Statement

Is it there a method to find out if a polynomial has no integer roots?

The Attempt at a Solution

I tried the division of polynomials, as well as the Horner's Method, but no luck.

 

[Kummer]

Homework Statement

Is it there a method to find out if a polynomial has no integer roots?

The Attempt at a Solution

I tried the division of polynomials, as well as the Horner's Method, but no luck.

Use "rational root theorem".

 

[ZioX]

Doesn't the rational root theorem only apply to polynomials over the rationals?

 

[Dick]

Yes, basically. Is there a method for general real polynomials?

 

[abelian jeff]

Povatix,

Does the polynomial itself have integer coefficients? If so, you can use Eisenstein's criterion.

 

[Dick]

Povatix,

Does the polynomial itself have integer coefficients? If so, you can use Eisenstein's criterion.

Irreducible is sufficient to show there is no rational root, but it's not necessary. Kummer was right. Just check all the possible rational roots, if none are integers, then there are none. The OP's polynomial almost certainly has rational coefficients. And if it doesn't I wish they had displayed it.

 

[povatix]

Here's the equation.

$$\chi^{10} + p\chi^{9} - q\chi^{7} + r\chi^{4} - s = 0$$

thnx in advance

 

[HallsofIvy]

If the coefficients are all integers, then the only possible integer solutions are s or its factors. The only way I know of determining whether any of those ARE solutions is to try them in the equation. Are you doing this for specific values of p, q, r, s? Obviously, the equation you give will have integer values for some values of p, q, r, s and not for others.

 

[povatix]

Oh right, sorry, the question also states that p q r and s are all odd integers

 

[HallsofIvy]

If x is an odd integer, then so is any power of it and so is any odd integer times any power. What can you say about the sum (or difference) of 5 odd integers?

If x is an even integer, then so is any power of it and so is any integer times any power. What can you say about the sum (or difference) of 4 even integers. In order that the value of the polynomial be 0, what must s be?

It would have helped a lot if you had given us all this information to begin with!

 

[Gib Z]

My thread here had the same problem: https://www.physicsforums.com/showthread.php?t=169364.

Im going to guess the OP lives in Australia and is doing the Math Enrichment program, Polya series.

 

[povatix]

Yes, it's true. This is part of a much larger enrichment program aimed at the top students in Year 10, which includes a textbook that is meant to aid you through the 16 questions.

P.S. How did you find out?

 

[Gib Z]

I'm Psychic =]

Jokes, I did it this year too. You will like that thread then, it has a lot of other questions from the program :)