Question: Find all solutions of the following polynomial equation. \[ -24 - 6x^3 + 32x + 10x^2 = 0 \]…

Find all solutions of the following polynomial equation.

\[ -24 - 6x^3 + 32x + 10x^2 = 0 \]

Answer

Separate multiple answers with commas.

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Solution

To solve the equation: \[ -24 - 6x^3 + 32x + 10x^2 = 0 \] Arrange the equation in standard form: \[ -6x^3 + 10x^2 + 32x - 24 = 0 \] Multiply both sides by -1 to simplify: \[ 6x^3 - 10x^2 - 32x + 24 = 0 \] Divide the entire equation by 2: \[ 3x^3 - 5x^2 - 16x + 12 = 0 \] Apply the Rational Root Theorem to identify possible rational roots: ±1, ±2, ±3, ±4, ±6, ±12, ±1/3, ±2/3, ±4/3. Test \( x = 3 \): \[ 3(3)^3 - 5(3)^2 - 16(3) + 12 = 81 - 45 - 48 + 12 = 0 \] Since \( x = 3 \) is a root, factor out \( (x - 3) \): \[ (x - 3)(3x^2 + 4x - 4) = 0 \] Solve the quadratic equation \( 3x^2 + 4x - 4 = 0 \): \[ x = \frac{-4 \pm \sqrt{16 + 48}}{6} = \frac{-4 \pm 8}{6} \] This gives: \[ x = \frac{4}{6} = \frac{2}{3}, \quad x = \frac{-12}{6} = -2 \] Solutions: \( 3, \frac{2}{3}, -2 \)