Divide \( :\left(2 x^{3}-3 x+1\right) \div(x-2) \)

To divide the polynomial \(2x^3 - 3x + 1\) by \(x - 2\), we can use polynomial long division. First, we set up the division: 1. Divide the leading term \(2x^3\) by \(x\) to get \(2x^2\). 2. Multiply \(2x^2\) by \(x - 2\) to get \(2x^3 - 4x\). 3. Subtract \(2x^3 - 4x\) from \(2x^3 - 3x + 1\) to get \(x + 1\). 4. Now, divide \(x\) by \(x\) to get \(1\). 5. Multiply \(1\) by \(x - 2\) to get \(x - 2\). 6. Subtract \(x - 2\) from \(x + 1\) to get \(3\). So, the result of the division is \(2x^2 + 1\) with a remainder \(3\). Hence, we can express the final answer as: \[ 2x^2 + 1 + \frac{3}{x - 2} \]