Proving inequality   $x^{10}-x^6+x^2-x+1>0$ How can the inequality  $x^{10}-x^6+x^2-x+1>0$ be proved
a) using elementary mathematical methods？
b) using higher mathematical methods？

Another elementary method:
We want to show that $x^{10}+x^2+1 >x^6+x$.  If we can show it for $x \ge 0$, it will be true always.  This is because if $x$ is positive, replacing $x$ by $-x$ does not change $x^{10}+x^2+1$, but turns  $x^6+x$ into the smaller $x^6-x$.
We now take care of  $x \ge 0$.  If $x\ge 1$, then $x^{10}\ge x^6$ and $x^2 \ge x$, so 
$x^{10}+x^2+1 >x^6+x$.
If $0 \le x < 1$, then $x^2 \ge x^6$ and $1 >x$, so $x^{10}+x^2+1 >x^6+x$.
