Divide. If the polynomial does not divide evenly, include the remainder as a fraction. \( \left(9 m^{5}-46 m^{4}-58 m^{3}+62 m^{2}-16 m+28\right) \div(m-6) \) Submit

To divide the polynomial \(9 m^{5}-46 m^{4}-58 m^{3}+62 m^{2}-16 m+28\) by \(m-6\), we can use synthetic division or long division. Here, we’ll apply synthetic division: 1. **Set up the synthetic division** using 6 (since we are dividing by \(m - 6\)): \[ \begin{array}{r|rrrrrr} 6 & 9 & -46 & -58 & 62 & -16 & 28 \\ & & 54 & 48 & -60 & 12 & -24 \\ \hline & 9 & 8 & -10 & 2 & -4 & 4 \\ \end{array} \] 2. **Perform the synthetic division** step by step: - Bring down the first coefficient (9). - Multiply 6 by 9 to get 54, add to -46 to get 8. - Multiply 6 by 8 to get 48, add to -58 to get -10. - Multiply 6 by -10 to get -60, add to 62 to get 2. - Multiply 6 by 2 to get 12, add to -16 to get -4. - Multiply 6 by -4 to get -24, add to 28 to get 4. 3. The final row shows the coefficients of the quotient polynomial and the remainder: - Quotient: \(9m^4 + 8m^3 - 10m^2 + 2m - 4\) - Remainder: 4 Thus, we can express the result of the division as: \[ \frac{9m^5 - 46m^4 - 58m^3 + 62m^2 - 16m + 28}{m - 6} = 9m^4 + 8m^3 - 10m^2 + 2m - 4 + \frac{4}{m - 6} \]