Solve the polynomial equation by factoring and check the solutions graphically.

Question

x cubed minus 16 x equals 0
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Part 1
The solutions are
  
enter your response here.
​(Simplify your answer. Use a comma to separate answers as​ needed.)

Answer 1

To solve the polynomial equation \( x^3 - 16x = 0 \) by factoring, we can start by factoring out the common term \( x \):

\[ x(x^2 - 16) = 0 \]

Next, we notice that the expression \( x^2 - 16 \) is a difference of squares, which can be factored further:

\[ x(x - 4)(x + 4) = 0 \]

Now, we set each factor equal to zero to find the solutions:

1. \( x = 0 \)
2. \( x - 4 = 0 \) \(\Rightarrow x = 4\)
3. \( x + 4 = 0 \) \(\Rightarrow x = -4\)

Thus, the solutions are:

\[ x = 0, 4, -4 \]

Checking Graphically

To check the solutions graphically, you would typically plot the function \( y = x^3 - 16x \) and look for the points where the graph intersects the x-axis. These intersections correspond to the solutions found above.

Using a graphing tool or software, you can visualize the function. The graph should show intersections at \( x = 0 \), \( x = 4 \), and \( x = -4 \).

The final answer is:

\[ \boxed{0, 4, -4} \]