$(4$ points$)$ Prove by contradiction that the equation

$x^3+x+1=0$

has no rational roots.

Hint: The approach to this problem is very similar to our proof that $\sqrt[\phantom{2}]{2}$ is irrational. Assume that a rational root exists, substitute that root back into the equation, and see what you can conclude about the parity of the variables. Use the results from the previous problem about the equation $a^3+a{b}^2+b^3=0.($Even if you didn't complete the previous problem, you can still use those results in this proof.$)$