MHB Proving that solutions of a equation are irrational

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The discussion focuses on proving that the solutions to the equation x^3 + x + 1 = 0 are irrational. The Rational Root Theorem is referenced, indicating that since neither x = -1 nor x = 1 are solutions, there are no rational roots. A proof by contradiction is suggested, starting with the assumption that x is rational and can be expressed as p/q, leading to a derived equation that reveals a contradiction. The approach parallels the proof of the irrationality of √2. Ultimately, the discussion emphasizes the use of algebraic manipulation to demonstrate the irrational nature of the solutions.
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how to prove that solutions of the following equation are irrational

x^3 + x + 1 = 0
 
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ranga519 said:
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.
 
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?
 

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