Question: Find all solutions of the following polynomial equation. \[ 4 - 2x^2 + x^3 - 2x = 0 \] Answer:…

Find all solutions of the following polynomial equation.

\[ 4 - 2x^2 + x^3 - 2x = 0 \]

Answer: 8 Points

Separate multiple answers with commas.

\[ (2, \sqrt{2}, -\sqrt{2}) \]

Solution

Let’s solve the polynomial equation step by step. First, consider the given equation: \[ 4 - 2x^2 + x^3 - 2x = 0 \] Rearrange the equation in standard form: \[ x^3 - 2x^2 - 2x + 4 = 0 \] Next, factor the polynomial by grouping: \[ x^3 - 2x^2 - 2x + 4 = (x^3 - 2x^2) + (-2x + 4) = x^2(x - 2) -2(x - 2) = (x^2 - 2)(x - 2) = 0 \] Set each factor equal to zero and solve for \( x \): \[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \] \[ x^2 - 2 = 0 \quad \Rightarrow \quad x = \sqrt{2}, \quad x = -\sqrt{2} \] Therefore, the solutions to the equation are: \[ 2, \sqrt{2}, -\sqrt{2} \]