A quesion on establishment of nature of roots

[Akshay_Anti]

we are given an equation x⁵+x=10
. How to prove that the only root for the equation is irrational? I'm an average 12th standard student. So, please keep it low. Thanks in advance.

 

[Infinitum]

Did you try using the Rational root Theorem?

It's quite clear that there is only one real root to the equation. If you can find a contradiction that the root cannot be rational, you are done

 

[Akshay_Anti]

How to contradict? Assuming a p/q form doesn't help... the root must be of form non- terminating non-repeating decimal expansion... how to do that?

 

[Number Nine]

How to contradict? Assuming a p/q form doesn't help... the root must be of form non- terminating non-repeating decimal expansion... how to do that?

The link Infinitum posted outlines the procedure. The rational zero theorem clearly describes all possible rational roots of the equation.