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.

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Solution

To solve the polynomial equation \[ 4 - 2x^2 + x^3 - 2x = 0 \] First, rearrange it in standard form: \[ x^3 - 2x^2 - 2x + 4 = 0 \] Look for possible rational roots using the Rational Root Theorem. The possible rational roots could be factors of the constant term (4) divided by factors of the leading coefficient (1): Possible rational roots: \( \pm 1, \pm 2, \pm 4 \) Test these roots: Substitute \( x = 1 \) into the polynomial: \[ 1^3 - 2(1)^2 - 2(1) + 4 = 1 - 2 - 2 + 4 = 1 \] \((x = 1)\) is not a root. Substitute \( x = -1 \): \[ (-1)^3 - 2(-1)^2 - 2(-1) + 4 = -1 - 2 + 2 + 4 = 3 \] \((x = -1)\) is not a root. Substitute \( x = 2 \): \[ 2^3 - 2(2)^2 - 2(2) + 4 = 8 - 8 - 4 + 4 = 0 \] \((x = 2)\) is a root. Now perform synthetic division using \( x = 2 \): \[ \begin{array}{r|rrrr} 2 & 1 & -2 & -2 & 4 \\ & & 2 & 0 & -4 \\ \hline & 1 & 0 & -2 & 0 \\ \end{array} \] The quotient is \( x^2 - 2 \). Set the quotient equal to zero: \[ x^2 - 2 = 0 \] Solve for \( x \): \[ x^2 = 2 \] \[ x = \pm \sqrt{2} \] The solutions for the polynomial are: \( x = 2, \sqrt{2}, -\sqrt{2} \)