# Polynomial with integer roots

1. Aug 6, 2007

### Klaus_Hoffmann

Hi,.. using a Sturm or other sequence, could we find how many integer roots have the Polynomial

$$K(x)= \sum_{n=0}^{d} a_{n}x^{n}$$

where all the 'a_n' are integers (either positive or negative)

2. Aug 6, 2007

### mathman

Sturm sequence will tell you the number of REAL roots greater than a preassigned number. However, it does not specifically single out integers.

With some effort you could used it by counting roots > n-e and roots >n+e (where e is small). If the difference is 1, then n (or something close to it) is a root.

3. Aug 7, 2007

### HallsofIvy

Staff Emeritus
This may not be what you are looking for but you can start with the rational root theorem: All rational roots have numerators that evenly divide a0 and denominators that evenly divide ad. Integer roots are rational roots with denominators equal to 1 so all possible integer roots must divide a0. Once you have determined all possible integer roots you will have to try them in the equation to see if they really are roots.

Last edited: Aug 7, 2007
4. Aug 7, 2007

### Gib Z

That sounds like that is what Klaus is exactly looking for
I've read this thread before..why didn't I think of that >.<..

5. Aug 7, 2007

### HallsofIvy

Staff Emeritus
I thought it was a little too simple- and it doesn't determine the number of integer roots, it helps you determine those roots so you can then count them!

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