# Polynomials do or don't have integer roots?

1. Aug 14, 2007

### povatix

1. The problem statement, all variables and given/known data

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

3. The attempt at a solution
I tried the division of polynomials, as well as the Horner's Method, but no luck.

2. Aug 14, 2007

### Kummer

Use "rational root theorem".

3. Aug 14, 2007

### ZioX

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

4. Aug 14, 2007

### Dick

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

5. Aug 14, 2007

### abelian jeff

Povatix,

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

6. Aug 14, 2007

### Dick

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.

7. Aug 20, 2007

### povatix

Here's the equation.

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

8. Aug 20, 2007

### HallsofIvy

Staff Emeritus
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.

9. Aug 20, 2007

### povatix

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

10. Aug 20, 2007

### HallsofIvy

Staff Emeritus
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!

11. Aug 21, 2007

### Gib Z

12. Aug 21, 2007

### 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?

13. Aug 21, 2007

### Gib Z

I'm Psychic =]

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