To solve the problem of dividing the polynomial \(4x^3 - 3x + 1\) by \(2x - 1\), we will use polynomial long division. Let's go through the steps in detail. ### Step-by-Step Explanation 1. **Set Up the Division**: We write \(4x^3 - 3x + 1\) (the dividend) under the long division symbol and \(2x - 1\) (the divisor) outside. \[ \begin{array}{r|l} 2x - 1 & 4x^3 + 0x^2 - 3x + 1 \\ \end{array} \] Note: We include \(0x^2\) to represent the missing \(x^2\) term. 2. **Divide the Leading Terms**: Divide the leading term of the dividend \(4x^3\) by the leading term of the divisor \(2x\): \[ \frac{4x^3}{2x} = 2x^2 \] This is the first term of our quotient. 3. **Multiply and Subtract**: Multiply \(2x^2\) by the entire divisor \(2x - 1\): \[ 2x^2(2x - 1) = 4x^3 - 2x^2 \] Now, subtract this from the original polynomial: \[ (4x^3 - 3x + 1) - (4x^3 - 2x^2) = 2x^2 - 3x + 1 \] 4. **Repeat the Process**: Now, we repeat the process with the new polynomial \(2x^2 - 3x + 1\). - Divide the leading term \(2x^2\) by \(2x\): \[ \frac{2x^2}{2x} = x \] - Multiply \(x\) by the divisor: \[ x(2x - 1) = 2x^2 - x \] - Subtract: \[ (2x^2 - 3x + 1) - (2x^2 - x) = -2x + 1 \] 5. **Final Division**: Now, we divide \(-2x + 1\) by \(2x - 1\). - Divide the leading term \(-2x\) by \(2x\): \[ \frac{-2x}{2x} = -1 \] - Multiply \(-1\) by the divisor: \[ -1(2x - 1) = -2x + 1 \] - Subtract: \[ (-2x + 1) - (-2x + 1) = 0 \] 6. **Conclusion**: The division is complete, and we have no remainder. The quotient is: \[ 2x^2 + x - 1 \] ### Final Answer The correct option is **D. \(2x^2 + x - 1\)**. ### Explanation of Other Options - **A. \(2x^2 - x + 1\)**: This option has the wrong sign for the \(x\) term and the constant term is incorrect. - **B. \(2x^2 - x - 1\)**: This option has the wrong sign for the constant term. - **C. \(2x^2 + x + 1\)**: This option has the wrong sign for the constant term. ### Revision Summary - Use polynomial long division to divide polynomials. - Always align like terms and subtract carefully. - Keep track of signs when multiplying and subtracting. - The final quotient is the result of the division, with no remainder in this case.