9. [-/1 Points] DETAILS MY NOTES TGEIALG6 5.8.091. Perform the division. (Simplify your answer completely.) \[ a - 5 \longdiv { a ^ { 2 } - 1 7 a + 6 6 } \] \( \square \)

To perform the division of the polynomial \( a^2 - 17a + 66 \) by \( a - 5 \), follow these steps: ### Step 1: Setup the Division \[ \begin{array}{r|rrr} a - 5 & a^2 & -17a & +66 \\ \hline & & & \\ \end{array} \] ### Step 2: Divide the Leading Terms - **Divide \( a^2 \) by \( a \)** to get \( a \). - **Multiply \( a \) by \( a - 5 \)** to get \( a^2 - 5a \). - **Subtract** this from the original polynomial: \[ \begin{array}{r|rrr} a - 5 & a^2 & -17a & +66 \\ \hline & -(a^2 - 5a) & & \\ \hline & 0 & -12a & +66 \\ \end{array} \] ### Step 3: Repeat the Process - **Divide \( -12a \) by \( a \)** to get \( -12 \). - **Multiply \( -12 \) by \( a - 5 \)** to get \( -12a + 60 \). - **Subtract** this from the current polynomial: \[ \begin{array}{r|rrr} a - 5 & a^2 & -17a & +66 \\ \hline & a - 12 & -(a^2 - 5a) & -( -12a + 60) \\ \hline & 0 & 0 & +6 \\ \end{array} \] ### Step 4: Write the Final Answer The division yields a quotient of \( a - 12 \) with a remainder of \( 6 \). Therefore, the simplified form is: \[ a - 12 + \frac{6}{a - 5} \] **Final Answer:** \( a - 12 + \dfrac{6}{\,a - 5\,} \)