Divide. If the polynomial does not divide evenly, include the remainder as a fraction. \( \left.\qquad 17 r^{3}+135 r^{2}+71\right) \div(r+8) \) Submit

To divide the polynomial \( 17r^3 + 135r^2 + 71 \) by \( r + 8 \), let's perform polynomial long division. 1. Divide the leading term: \( 17r^3 \div r = 17r^2 \). 2. Multiply: \( 17r^2 \cdot (r + 8) = 17r^3 + 136r^2 \). 3. Subtract the result from the original polynomial: \[(17r^3 + 135r^2 + 71) - (17r^3 + 136r^2) = -r^2 + 71.\] 4. Now, divide the leading term of the new polynomial: \(-r^2 \div r = -r\). 5. Multiply: \(-r \cdot (r + 8) = -r^2 - 8r\). 6. Subtract again: \[(-r^2 + 71) - (-r^2 - 8r) = 8r + 71.\] 7. Now, divide the leading term of the new polynomial: \(8r \div r = 8\). 8. Multiply: \(8 \cdot (r + 8) = 8r + 64\). 9. Subtract once more: \[(8r + 71) - (8r + 64) = 7.\] So, the complete division yields: \[ 17r^2 - r + 8 + \frac{7}{r + 8}. \] Thus, the result is \( 17r^2 - r + 8 + \frac{7}{r + 8} \).