# Polynomial Algebra: Show Alpha is Power of Prime p

• Coolster7
In summary, if a polynomial has a positive power of a prime number as its constant term, then any integer that makes the polynomial equal to zero must also be a power of that prime number. This can be shown using the rational root theorem and factoring.
Coolster7

## Homework Statement

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.

## 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:
Do you know the rational root theorem?

Coolster7 said:
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.

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

1 person
kduna said:
You are correct so far. If you factor out alpha, then you will have that alpha divides ##p^n##.

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

Coolster7 said:
Thanks for your help. So because alpha divides p^n this means alpha is also a power of p I'm assuming.

Yep!

1 person

## 1. What is a polynomial?

A polynomial is an algebraic expression that consists of variables, coefficients, and exponents. It can contain addition, subtraction, and multiplication operations, but not division.

## 2. What is the degree of a polynomial?

The degree of a polynomial is the highest exponent of the variable in the expression. For example, in the polynomial 3x^2 + 5x + 2, the degree is 2 because 2 is the highest exponent of x.

## 3. How do you determine if a polynomial is a power of a prime number?

A polynomial is a power of a prime number if all of its terms have the same degree and the coefficients are all multiples of the prime number. Additionally, the coefficient of the term with the highest degree must be equal to the prime number raised to that degree.

## 4. What is Alpha in polynomial algebra?

In polynomial algebra, Alpha is a variable that represents any number or expression. It is commonly used to denote the roots or solutions of a polynomial equation.

## 5. How can you show that Alpha is a power of prime p?

To show that Alpha is a power of prime p, you can substitute Alpha into the polynomial expression and simplify it to the form of p^n, where n is a positive integer. This will demonstrate that Alpha is a multiple of p to the power of n and therefore a power of prime p.

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