Solution to the cubic equation

Nov 2011
Let a,b be positive real numbers. Prove there there exists a unique solution to x[SUP]3[/SUP]+ax=b.

I can prove that if there is a solution it is unique, but I'm not sure how to prove that a solution definitely exists.

Any help would be great.

Aug 2008
All polynomials with real coefficients have as many complex roots as the degree of the polynomial. All nonreal roots appear as complex conjugates, so if the polynomial is of odd degree, there has to be at least one real root as it can't have a conjugate pair.
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