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

Question

x cubed minus 4 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 \(x^3 - 4x = 0\) by factoring, we can start by factoring out the common term.

1. The equation can be rewritten as: \[ x^3 - 4x = 0 \] Factoring out \(x\), we have: \[ x(x^2 - 4) = 0 \]

2. Next, we further factor \(x^2 - 4\) as it is a difference of squares: \[ x(x - 2)(x + 2) = 0 \]

3. Setting each factor to zero gives us the solutions:

   • \(x = 0\)
   • \(x - 2 = 0 \Rightarrow x = 2\)
   • \(x + 2 = 0 \Rightarrow x = -2\)

Thus, the solutions to the equation \(x^3 - 4x = 0\) are: \[ x = 0, 2, -2 \]

Part 1 Answer: The solutions are \(0, 2, -2\).

Graphical Check

To check the solutions graphically, you can plot the function \(y = x^3 - 4x\) and observe where it intersects the x-axis. The intersections with the x-axis correspond to the solutions of the equation.

When plotted, the graph of the function will cross the x-axis at \(x = -2\), \(x = 0\), and \(x = 2\), confirming our solutions.