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

Find all solutions of the following polynomial equation.

$$-6 + x^3 - x + 6x^2 = 0$$

Answer

Separate multiple answers with commas.

Solution

To solve the polynomial equation: \[ -6 + x^3 - x + 6x^2 = 0 \] First, rearrange the terms: \[ x^3 + 6x^2 - x - 6 = 0 \] Next, identify possible rational roots using the Rational Root Theorem. The possible roots are factors of the constant term divided by factors of the leading coefficient, which are: \[ \pm 1, \pm 2, \pm 3, \pm 6 \] Testing \( x = 1 \): \[ 1^3 + 6(1)^2 - 1 - 6 = 1 + 6 - 1 - 6 = 0 \] Since \( x = 1 \) is a root, factor out \( (x - 1) \): \[ (x - 1)(x^2 + 7x + 6) = 0 \] Now, solve the quadratic equation \( x^2 + 7x + 6 = 0 \): \[ \begin{cases} x + 1 = 0 & \\ x + 6 = 0 & \end{cases} \] So, the solutions are: \[ x = 1, -1, -6 \]