For which values $a,b$ $\in$ $\mathbb{R}$ exists a double zero of $f(x) = x^3-ax+b$? For which values $a,b$ exist exactly one, two or three real zeros of $f$, respectively?

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I'm not sure how to approach this, I only found out that for $a=0$ and $b=0$ both $f$ and $f^`$ have a zero so I guess I found the double zero. But how do I determine adequate values for $a$ and $b$ to get a specific number of zeros?
I thought about using the Newton method somehow but I'm kind of lost at the moment. 
All help is very much appeciated, so thanks in advance.