What's the Second Part?

Date: 21 Mar 1995 01:29:45 -0500
From: peter eckersley
Subject: help

The question:

If x = root3 + root2, show that x^4 - 10x^2 + 1 = 0.
Hence prove that root3 + root2 is irrational.

The first part is ok by substitution.
The second part... ?

Peter Eckersley
Valley View Secondary School

Date: 21 Mar 1995 10:11:13 -0500
From: Dr. Ken
Subject: Re: help

Hello there!

Well, here are two things I'd tell you about this:

a) Use the quadratic formula on the quartic equation, and use 
x^2 as your variable.  So you get x^2 = (10 +- Sqrt{100 - 4})/2, 
and then simplify.  What does this information about x^2 tell 
you about x?

b) I wonder whether the person writing the question really meant 
to say "Hence prove that root3 + root2 is algebraic."  Because if 
you can find a polynomial that has the number in question as a 
root, you've just satisfied the definition of algebraic.

I hope this helps!

-Ken "Dr." Math