How to show $x^{10}-2x^9+3x^8-...-10x+11=0$ has no real root? I tried to solve below equation $$x^{10}-2x^9+3x^8-...-10x+11=0$$ I plot the graph and see there is no real root , But I get stuck how to show analytical .Can some one help me or give an idea ? 
Thanks in advance.
This is graph of the function $$f(x)=x^{10}-2x^9+3x^8-...-10x+11$$
https://www.desmos.com/calculator/6k1lj498ra

There is some confusion over the sign of the coefficients. But it might prove useful to consider $$x^8(x-1)^2+2x^6(x-1)^2+3x^4(x-1)^2+4x^2(x-1)^2 =$$$$=x^{10}-2x^9+x^8+2x^8-4x^7+2x^6+3x^6-6x^5+3x^4+4x^4-8x^3+4x^2 =$$$$=x^{10}-2x^9+3x^8-4x^7+5x^6-6x^5+7x^4-8x^3+4x^2$$which is clearly non-negative, or some variant of this. You should be able to conclude from there.
