To solve the polynomial equation \( x^3 - 100x = 0 \), we can factor the expression.
First, we can factor out the common term \( x \):
\[ x(x^2 - 100) = 0 \]
Next, we can factor \( x^2 - 100 \) as it is a difference of squares:
\[ x^2 - 100 = (x - 10)(x + 10) \]
Now, the equation becomes:
\[ x(x - 10)(x + 10) = 0 \]
Setting each factor equal to zero gives us the solutions:
- \( x = 0 \)
- \( x - 10 = 0 \) ⟹ \( x = 10 \)
- \( x + 10 = 0 \) ⟹ \( x = -10 \)
Thus, the solutions are:
\[ \boxed{0, 10, -10} \]
To verify graphically, you can sketch the function \( y = x^3 - 100x \). The x-intercepts of the graph will indicate the points where the function equals zero, which should correspond to our solutions \( x = 0, 10, -10 \).