MHB Proving that solutions of a equation are irrational

[ranga519]

how to prove that solutions of the following equation are irrational

x^3 + x + 1 = 0

 

[I like Serena]

how to prove that solutions of the following equation are irrational

x^3 + x + 1 = 0

Hi Ranga,

I do not know what your course material says, but we have the Rational Root Theorem to work with.
Since neither $x=-1$ nor $x=1$ is a solution, that implies that there is no rational root.

 

[Pereskia]

You can prove it by contradiction. It is essentially the same proof as the proof of $\sqrt 2$ being irrational.

Assume x is rational, then x can be given as $x=p/q$ where p and q have no common primefactors. The equation becomes $$\frac{p^3}{q^3}+\frac{p}{q}+1=0 \Leftrightarrow p^3=-q^2(p+q)$$
Can you se the contradiction now?