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. Of the remaining four roots, at least two and at most four are complex (because complex roots come in pairs.) 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. |
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