# Polynomial Algebra

1. Feb 19, 2014

### Coolster7

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

Let f(x) = anxn + an-1xn-1 + .... + a1x + a0 be a polynomial where the coefficients an, an-1, ... , a1, a0 are integers.

Suppose a0 is a positive power of a prime number p.

Show that if $\alpha$ is an integer for which f( $\alpha$ ) = 0, $\alpha$ is also a power of p.

2. Relevant equations

3. The attempt at a solution

I substituted $\alpha$ into the equation in the place of x for each term. I also substituted in pn in the place of a0 as this is a positive power of a prime number p (as given in the question). This gave me:

f($\alpha$) = an$\alpha$n + an-1$\alpha$n-1 + .... + a1$\alpha$ + pn = 0

I then decided to isolate pn by moving the other terms to the other side of the equation which gave me:

pn = -{an$\alpha$n + an-1$\alpha$n-1 + .... + a1$\alpha$}

Is what I have done so far correct? I now have to show from this that $\alpha$ is also a power of p. I'm unsure what the next step is to do that.

Last edited by a moderator: Feb 20, 2014
2. Feb 19, 2014

### LCKurtz

Do you know the rational root theorem?

3. Feb 19, 2014

### kduna

You are correct so far. If you factor out alpha, then you will have that alpha divides $p^n$.

4. Feb 20, 2014

### Coolster7

Thanks for your help. So because alpha divides p^n this means alpha is also a power of p I'm assuming.

5. Feb 20, 2014

Yep!