Questions: Solve for x. -x^3-2 x^2+8 x=0 x=[?], ,

Solve for x. -x^3-2 x^2+8 x=0 x=[?], ,
Transcript text: Solve for x . \[ \begin{array}{c} -x^{3}-2 x^{2}+8 x=0 \\ x=[?], \square, \square \end{array} \]
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Solution

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Solution Steps

Step 1: Factor the Polynomial

The given polynomial equation is

\[ -x^{3}-2x^{2}+8x=0 \]

Factoring this polynomial yields:

\[ -x \left(x - 2\right) \left(x + 4\right) = 0 \]

Step 2: Solve for \( x \)

To find the values of \( x \), we set each factor equal to zero:

  1. \( -x = 0 \) ⟹ \( x = 0 \)
  2. \( x - 2 = 0 \) ⟹ \( x = 2 \)
  3. \( x + 4 = 0 \) ⟹ \( x = -4 \)
Step 3: Sort the Solutions

The solutions obtained are:

\[ x = -4, \quad x = 0, \quad x = 2 \]

Sorting these solutions gives:

\[ x = -4, \quad x = 0, \quad x = 2 \]

Final Answer

The smallest solution is

\[ \boxed{x = -4} \]

The complete set of solutions is

\[ \boxed{x = -4, 0, 2} \]

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