Finding Polynomials from their Zeros

[gabinchi]

Homework Statement

Prove that x = 2^(1/2)+2^(1/3) is irrational.

Homework Equations

The Attempt at a Solution

The hint to the exercise says that I should first show that x satisfies an equation of the form x^6+a_1*X^5+...+a_0 = 0, where the coefficients are integers.

I am having trouble showing that x satisfies such an equation.

I tried multiplying (x-2^(1/2)-2^(1/3))(x+2^(1/2)-2^(1/3)) but was not successful in getting integer coefficients.

 

[matt grime]

Try putting some more factors in. Or just start from

x=sqrt(2)+cbrt(2)

and manipulate it to find an expression x satsifies. (eg, if x=sqrt(2), then squaring both sides yields x^2=2, so x satisfies x^2-2=0. n.b. don't square both sides of what you started wit -try something more sophisticated than that).