<s>[INST] Why is the sum of the roots of $x^3+9=12x$ zero? I found the roots of the equation to be 
$$x=3,\frac{\sqrt{21}}{2}-\frac{3}{2},-\frac{\sqrt{21}}{2}-\frac{3}{2}$$
Algebraically, why is the sum of these roots equal to zero?
 [/INST] Consider a monic polynomial $f(x)$ of degree three with three roots: $a_0$, $a_1$, $a_2$. So
$$f(x) = (x - a_0)(x - a_1)(x - a_2).$$
Expanding gives
$$f(x) = x^3 - (a_0 + a_1 + a_2) x^2 + (\dots) x + (\dots).$$
So, the coefficient of $x^2$ is the negation of the sum of the roots.
In your case, the coefficient of $x^2$ is $0$, so the sum of the roots is $0$.
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