Question

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Part 1
Solve the polynomial equation by factoring and check the solutions graphically.
x cubed minus 100 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 \( x^3 - 100x = 0 \), 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:

1. \( x = 0 \)
2. \( x - 10 = 0 \) ⟹ \( x = 10 \)
3. \( x + 10 = 0 \) ⟹ \( x = -10 \)

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 \( x = 0, 10, -10 \).