My Math Forum  

Go Back   My Math Forum > High School Math Forum > Algebra

Algebra Pre-Algebra and Basic Algebra Math Forum

LinkBack Thread Tools Display Modes
September 28th, 2013, 08:14 AM   #1
Joined: Jun 2013

Posts: 31
Thanks: 0

Rational number

Let x be a real number with the property: x^3+x and x^5+x are rational. Prove, that x is rational.
tva_vlad is offline  
October 7th, 2013, 02:38 AM   #2
Joined: Mar 2013

Posts: 90
Thanks: 0

Re: Rational number

If is rational, must be of the form either
where are rational and is a positive non-square rational, or
where are rational and is a positive non-cube rational. Let us look at each of these cases.

Case 1:

We have

For this to be rational, the coefficient of must vanish. Hence (since ) and so is rational.

Case 2:

We have . Then

or .

This leads to and . If are not both zero, this means must take one of the following forms:



Take the first one.

Now examine the coefficient of in . It works out as . Then the coefficient of in is . Since this cannot be zero, is irrational. But if and are both rational then their difference must be rational as well. This contradiction shows that is impossible.

The other one leads to a similar impasse. Thus if , we must have and so which is rational.

Since is not a perfect cubic rational, this has no non-zero rational solution in so again and is rational.
Nehushtan is offline  

  My Math Forum > High School Math Forum > Algebra

number, rational

Thread Tools
Display Modes

Similar Threads
Thread Thread Starter Forum Replies Last Post
Find a rational number shunya Elementary Math 3 March 18th, 2014 09:37 AM
Rational number goodfeeling Algebra 17 October 26th, 2012 01:40 PM
Find rational number a and b such that: luketapis Algebra 8 June 22nd, 2012 01:18 AM
Rational Number Approximations guynamedluis Real Analysis 6 September 14th, 2011 12:52 AM
Prove a rational number??? samuraibob Algebra 5 August 3rd, 2010 12:45 AM

Copyright © 2018 My Math Forum. All rights reserved.