- #1

Coolster7

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## Homework Statement

Let f(x) = a

_{n}x

^{n}+ a

_{n-1}x

^{n-1}+ ... + a

_{1}x + a

_{0}be a polynomial where the coefficients a

_{n}, a

_{n-1}, ... , a

_{1}, a

_{0}are integers.

Suppose a

_{0}is a positive power of a prime number p.

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

## Homework Equations

## The Attempt at a Solution

I substituted [itex]\alpha[/itex] into the equation in the place of x for each term. I also substituted in p

^{n}in the place of a

_{0}as this is a positive power of a prime number p (as given in the question). This gave me:

f([itex]\alpha[/itex]) = a

_{n}[itex]\alpha[/itex]

^{n}+ a

_{n-1}[itex]\alpha[/itex]

^{n-1}+ ... + a

_{1}[itex]\alpha[/itex] + p

^{n}= 0

I then decided to isolate p

^{n}by moving the other terms to the other side of the equation which gave me:

p

^{n}= -{a

_{n}[itex]\alpha[/itex]

^{n}+ a

_{n-1}[itex]\alpha[/itex]

^{n-1}+ ... + a

_{1}[itex]\alpha[/itex]}

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

Can anyone help please?

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