Prove that $x^{16}-x^{11}+x^6-x+1>0$ for $x\in R$.
So I thought of something like this: $x^{10}(x^6-x)+x^6-x>-1$ $(x^{10}+1)(x^6-x)>-1$
But it seems to not be too much of help. While the first bracket is always positive, the latter can't be (or I don't know how to do this) easily transformed to a form showing that it's positive. I know it can be solved by thinking about three cases (for x>1, x<0 and $x\in <0,1>$), but it involves transforming the initial inequality three times each time basing rather on our skill with such problems (I mean - someone who hasn't worked on such previously would have a hard time getting the inequality to a form suitable for showing every case) rather than observations on-the-spot but can this be solved in some easier manner?