Prove $x^{4}-x+1=0$ has no solution 
I would like to prove that the following equation has no solution in $\mathbb{R}$
   $$x^{4}-x+1=0$$
my question : could we use  Intermediate Value Theorem to prove it otherways I'm interested in more ways of prove that has no solution in $\mathbb{R}$.
without :
i know that we can prove it by that way :
$x=x^4+1\geq 1$ and $x^4-x+1=x^2(x^2-1)+x^2-x+1>0$


If $x = x^{4}+1$, then certainly $x \geq 1$ as $x^{4} \geq 0$ for all real $x$. But then $x^{4}  = x^{3}x \geq x$ and $x^{4}+1 \geq x+1 > x$, a contradiction.
