Question: Find all solutions of the following polynomial equation. \[ -18 + 6x^2 + x^3 - 3x = 0 \] Answer…

Find all solutions of the following polynomial equation.

\[ -18 + 6x^2 + x^3 - 3x = 0 \]

Answer 8 Points

Separate multiple answers with commas.

Solution

To solve the polynomial equation: \[ x^3 + 6x^2 - 3x - 18 = 0 \] First, identify possible rational roots using the Rational Root Theorem. The possible roots are: \[ \pm1, \pm2, \pm3, \pm6, \pm9, \pm18 \] Test \( x = -6 \): \[ (-6)^3 + 6(-6)^2 - 3(-6) - 18 = -216 + 216 + 18 - 18 = 0 \] Since the equation equals zero, \( x = -6 \) is a root. Factor the polynomial using the root \( x = -6 \): \[ (x + 6)(x^2 - 3) = 0 \] Set each factor equal to zero: \[ x + 6 = 0 \quad \text{or} \quad x^2 - 3 = 0 \] Solve for \( x \): \[ x = -6 \] \[ x = \sqrt{3}, \quad x = -\sqrt{3} \] Therefore, the solutions are: \[ -6, \sqrt{3}, -\sqrt{3} \]