Prove that x is irrational unless it is an integer.

In summary, the bolded step in the problem is to show that if q does not divide pn, then x is an integer and q must be ±1.
  • #1
LaMantequilla
8
0

Homework Statement



This is taken from an answer book that I have. I don't understand the bolded step. Can someone explain it to me?Suppose x = p/q where p and q are natural numbers with no common factor. Then:

pn/qn + an-1pn-1/qn-1 + ... + ao = 0

and multiplying both sides by qn gives

pn + an-1pn-1q + ... + aoqn = 0

Now if q ≠ ±1 then q has some prime number as a factor. This prime number divides every term of the second equation other than pn, so it must divide pn also. Therefore it divides p, a contradiction. So q = ±1, which means that x is an integer.

Once again, it's the bolded step that I don't understand. Why must it divide pn?
Thanks in advance.
 
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  • #2
LaMantequilla said:

Homework Statement



This is taken from an answer book that I have. I don't understand the bolded step. Can someone explain it to me?

Suppose x = p/q where p and q are natural numbers with no common factor. Then:

pn/qn + an-1pn-1/qn-1 + ... + ao = 0

and multiplying both sides by qn gives

pn + an-1pn-1q + ... + aoqn = 0

Now if q ≠ ±1 then q has some prime number as a factor. This prime number divides every term of the second equation other than pn, so it must divide pn also. Therefore it divides p, a contradiction. So q = ±1, which means that x is an integer.

Once again, it's the bolded step that I don't understand. Why must it divide pn?
Thanks in advance.
Looks like you're asking us to pick things up in the middle of a proof, without letting us know what it is that is being proved.

I'm guessing that you're looking at a solution for the following.
Prove that any root of the following polynomial of degree, n, with integer coefficients:
xn + an-1 xn-1 + an-2 xn-2 + an-3 xn-3 + an-4 xn-4 + … + a0
is either an integer, or the root is irrational.​
The proof is by contradiction, and done by assuming that there is a rational, non-integer root.To answer your question:

Rewrite that second equation of yours as:

an-1pn-1q + ... + aoqn = -pn

So, q divides the left hand side. Therefore, it must divide the right hand side.
 
  • #3
Thank you so much! I can't believe I missed that! Thanks!
 

1. What does it mean for a number to be irrational?

A number is considered irrational if it cannot be expressed as a ratio of two integers. This means that its decimal representation is non-repeating and non-terminating.

2. Why is it important to prove that x is irrational unless it is an integer?

Proving that a number is irrational unless it is an integer is important because it helps us understand the properties of real numbers and their relationship to rational and irrational numbers. It also allows us to make mathematical statements and proofs with greater accuracy and precision.

3. What is the process for proving that a number is irrational?

To prove that a number is irrational, we can use a proof by contradiction. This means assuming that the number is rational and then showing that this assumption leads to a contradiction. This contradiction then proves that the number must be irrational.

4. Can any number be proven to be irrational?

No, not all numbers can be proven to be irrational. Some numbers, like integers and fractions, are considered rational and can be expressed as a ratio of two integers. Irrational numbers, on the other hand, cannot be expressed in this way and thus cannot be proven to be irrational using this method.

5. What are some examples of numbers that are irrational unless they are integers?

Some examples of numbers that are irrational unless they are integers include pi (3.14159...), e (2.71828...), and the square root of 2 (√2 = 1.41421...). These numbers cannot be expressed as a ratio of two integers and have decimal representations that are non-repeating and non-terminating.

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