Solve the polynomial equation by factoring, and check the solutions graphically.
\[
x^{3}-9 x=0
\]

Rewrite the equation in a completely factored form.
\[
x(x-3)(x+3)=0
\]
(Type your answer in factored form.)
The solution set is $\{\square\}$.
(Simplify your answer. Use a comma to separate answers as needed.)

Step 1 ：Given the polynomial equation \(x^{3}-9 x=0\).

Step 2 ：Rewrite the equation in a completely factored form: \(x(x-3)(x+3)=0\).

Step 3 ：The roots of the equation are the solutions to the equation. Set each factor equal to zero and solve for x: \(x=0\), \(x-3=0\), \(x+3=0\).

Step 4 ：Solving these gives the solutions \(x=-3\), \(x=0\), \(x=3\).

Step 5 ：Check the solutions graphically by plotting the equation and observing where it intersects the x-axis. The points of intersection are the solutions to the equation.

Step 6 ：The graph of the equation \(y = x^3 - 9x\) intersects the x-axis at -3, 0, and 3, which are the solutions we found earlier. This confirms that the solutions are correct.

Step 7 ：Final Answer: The solution set is \(\boxed{-3, 0, 3}\).