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\begin{tabular}{|l} I can divide polynomials using the box method. \\ Divide using the long division (box method). MUST SHOW WORK FOR FULL CREDIT. \\ \( \left(4 x^{3}+x-2\right) \div(2 x-1) \) \\ \hline\end{tabular}

Ask by Ortiz Robbins. in the United States
Mar 18,2025

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The division of \(4x^3 + x - 2\) by \(2x - 1\) using the long division method results in: \[ 2x^2 + x + 1 - \frac{1}{2x - 1} \]

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To divide the polynomial \(4x^3 + x - 2\) by \(2x - 1\) using the box method (also known as polynomial long division), we can set it up as follows: 1. **Set up the long division**: Write \(4x^3 + 0x^2 + x - 2\) under the division symbol. Remember to include a \(0x^2\) term so all degrees are represented. 2. **Divide the leading term**: Divide the leading term of the dividend \(4x^3\) by the leading term of the divisor \(2x\) to get \(2x^2\). 3. **Multiply and subtract**: Multiply \(2x^2\) by \(2x - 1\) to get \(4x^3 - 2x^2\). Subtract this from the original polynomial: \[ (4x^3 + 0x^2 + x - 2) - (4x^3 - 2x^2) = 0 + 2x^2 + x - 2. \] 4. **Repeat the process**: Now bring down \(2x^2 + x - 2\) and repeat the steps: - Divide \(2x^2\) by \(2x\) to get \(x\). - Multiply \(x\) by \(2x - 1\) to get \(2x^2 - x\). - Subtract: \[ (2x^2 + x - 2) - (2x^2 - x) = 0 + 2x - 2. \] 5. **Final division**: Now bring down \(2x - 2\): - Divide \(2x\) by \(2x\) to get \(1\). - Multiply \(1\) by \(2x - 1\) to get \(2x - 1\). - Subtract: \[ (2x - 2) - (2x - 1) = -1. \] 6. **Conclusion**: The remainder is \(-1\), and the complete result of the division is: \[ 2x^2 + x + 1 - \frac{1}{2x - 1}. \] So the answer is: \[ \frac{4x^3 + x - 2}{2x - 1} = 2x^2 + x + 1 - \frac{1}{2x - 1}. \]

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