## Solving the Polynomial Equation: 6x^3 + x^2 - 3x(x-1) + 2 = 0

This article explores the process of solving the polynomial equation **6x^3 + x^2 - 3x(x-1) + 2 = 0**. We will use algebraic manipulation to simplify the equation and then apply appropriate methods to find its solutions.

### Step 1: Simplifying the Equation

First, we need to simplify the equation by expanding the products and combining like terms:

**Expand the product:**-3x(x-1) = -3x^2 + 3x**Combine like terms:**6x^3 + x^2 - 3x^2 + 3x + 2 = 0**Simplify:**6x^3 - 2x^2 + 3x + 2 = 0

Now we have a simplified cubic equation.

### Step 2: Finding Solutions

Solving cubic equations can be complex. Here are a few common methods:

**Factoring:**Try factoring the equation. This might require using techniques like grouping or the rational root theorem.**Rational Root Theorem:**The rational root theorem can help find potential rational roots of the equation.**Numerical Methods:**If factoring proves difficult, numerical methods like Newton-Raphson iteration can be used to approximate the solutions.

### Step 3: Applying a Method

For this specific equation, factoring might be the most efficient method. Here's how we can approach it:

**Look for common factors:**There are no common factors among all the terms.**Grouping:**We can group the terms as follows: (6x^3 - 2x^2) + (3x + 2) = 0**Factor out common factors:**2x^2(3x - 1) + 1(3x + 2) = 0

Unfortunately, further factoring this expression proves difficult. Therefore, we will explore numerical methods to approximate the solutions.

### Step 4: Numerical Methods (Optional)

Using numerical methods like Newton-Raphson iteration requires an initial guess and an iterative process. This can be done using a calculator or specialized software.

### Conclusion

Solving the polynomial equation 6x^3 + x^2 - 3x(x-1) + 2 = 0 involves simplifying the equation, attempting factoring, and potentially employing numerical methods if factoring proves unsuccessful. The specific approach depends on the complexity of the equation and available resources.