# Find the number of real roots of the polynomial

Find the number of real roots of the polynomial $$f(x)=x^5+x^3-2x+1$$ If I use Descarte's Rule then I get $$f(x)=x^5+x^3-2x+1$$ there can't be more than two positive real roots. Again $$f(-x)=-x^5-x^3+2x+1$$ there can't be more than one negative real root.

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It's obvious there is one negative real root, because the graph goes through $(0,1)$, and the end behavior of the graph is down on the left hand side.
The derivative shows that the only positive real local minimum occurs at $x=\sqrt{0.4}$, but checking this in the equation shows that it is still above the $x$ axis. So the graph does not cross the x-axis on the positive real numbers, and the remaining four roots are complex.