Use the long division method to find the result when \( 6 x^{3}-11 x^{2}-27 x+10 \) is divided \( 3 x-1 \).

To divide \( 6x^{3} - 11x^{2} - 27x + 10 \) by \( 3x - 1 \) using the long division method, follow these steps: ### Step 1: Setup the Division \[ \begin{array}{r|llll} 3x - 1 & 6x^{3} & -11x^{2} & -27x & +10 \\ \end{array} \] ### Step 2: Divide the Leading Terms - **Divide:** \( 6x^{3} \div 3x = 2x^{2} \) - **Write:** \( 2x^{2} \) above the division line. \[ \begin{array}{r|llll} & 2x^{2} & & & \\ \hline 3x - 1 & 6x^{3} & -11x^{2} & -27x & +10 \\ \end{array} \] ### Step 3: Multiply and Subtract - **Multiply:** \( 2x^{2} \times (3x - 1) = 6x^{3} - 2x^{2} \) - **Subtract:** \[ \begin{align*} (6x^{3} - 11x^{2}) - (6x^{3} - 2x^{2}) &= -9x^{2} \end{align*} \] - Bring down the next term: \( -27x \). \[ \begin{array}{r|llll} & 2x^{2} & & & \\ \hline 3x - 1 & 6x^{3} & -11x^{2} & -27x & +10 \\ & -(6x^{3} - 2x^{2}) & & & \\ \hline & & -9x^{2} & -27x & \\ \end{array} \] ### Step 4: Repeat the Process - **Divide:** \( -9x^{2} \div 3x = -3x \) - **Write:** \( -3x \) above the division line. \[ \begin{array}{r|llll} & 2x^{2} & -3x & & \\ \hline 3x - 1 & 6x^{3} & -11x^{2} & -27x & +10 \\ & -(6x^{3} - 2x^{2}) & & & \\ \hline & & -9x^{2} & -27x & \\ & & -(-9x^{2} + 3x) & & \\ \hline & & & -30x & \\ \end{array} \] - **Multiply:** \( -3x \times (3x - 1) = -9x^{2} + 3x \) - **Subtract:** \[ \begin{align*} (-9x^{2} - 27x) - (-9x^{2} + 3x) &= -30x \end{align*} \] - Bring down the next term: \( +10 \). ### Step 5: Final Division - **Divide:** \( -30x \div 3x = -10 \) - **Write:** \( -10 \) above the division line. \[ \begin{array}{r|llll} & 2x^{2} & -3x & -10 & \\ \hline 3x - 1 & 6x^{3} & -11x^{2} & -27x & +10 \\ & -(6x^{3} - 2x^{2}) & & & \\ \hline & & -9x^{2} & -27x & \\ & & -(-9x^{2} + 3x) & & \\ \hline & & & -30x & +10 \\ & & & -(-30x + 10) & \\ \hline & & & & 0 \\ \end{array} \] - **Multiply:** \( -10 \times (3x - 1) = -30x + 10 \) - **Subtract:** \[ \begin{align*} (-30x + 10) - (-30x + 10) &= 0 \end{align*} \] ### Conclusion The division results in: \[ \boxed{2x^{2} - 3x - 10} \] There is **no remainder**.