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

−x3−2x2+8x=0 -x^{3}-2x^{2}+8x=0

Factoring this polynomial yields:

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

Step 2: Solve for x x

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

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

The solutions obtained are:

x=−4,x=0,x=2 x = -4, \quad x = 0, \quad x = 2

Sorting these solutions gives:

x=−4,x=0,x=2 x = -4, \quad x = 0, \quad x = 2

Final Answer

The smallest solution is

x=−4 \boxed{x = -4}

The complete set of solutions is

x=−4,0,2 \boxed{x = -4, 0, 2}

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