Divide using synthetic division $$ ( 11x^{6}-9x^{5}-17x^{3}+27x^{2}-13x+39 ) \div ( x-1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}1&11&-9&0&-17&27&-13&39\\& & 11& 2& 2& -15& 12& \color{black}{-1} \\ \hline &\color{blue}{11}&\color{blue}{2}&\color{blue}{2}&\color{blue}{-15}&\color{blue}{12}&\color{blue}{-1}&\color{orangered}{38} \end{array} $$The solution is:
$$ \frac{ 11x^{6}-9x^{5}-17x^{3}+27x^{2}-13x+39 }{ x-1 } = \color{blue}{11x^{5}+2x^{4}+2x^{3}-15x^{2}+12x-1} ~+~ \frac{ \color{red}{ 38 } }{ x-1 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -1 = 0 $ ( $ x = \color{blue}{ 1 } $ ) at the left.
$$ \begin{array}{c|rrrrrrr}\color{blue}{1}&11&-9&0&-17&27&-13&39\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}1&\color{orangered}{ 11 }&-9&0&-17&27&-13&39\\& & & & & & & \\ \hline &\color{orangered}{11}&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 11 } = \color{blue}{ 11 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{1}&11&-9&0&-17&27&-13&39\\& & \color{blue}{11} & & & & & \\ \hline &\color{blue}{11}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ 11 } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrrrrr}1&11&\color{orangered}{ -9 }&0&-17&27&-13&39\\& & \color{orangered}{11} & & & & & \\ \hline &11&\color{orangered}{2}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 2 } = \color{blue}{ 2 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{1}&11&-9&0&-17&27&-13&39\\& & 11& \color{blue}{2} & & & & \\ \hline &11&\color{blue}{2}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 2 } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrrrrr}1&11&-9&\color{orangered}{ 0 }&-17&27&-13&39\\& & 11& \color{orangered}{2} & & & & \\ \hline &11&2&\color{orangered}{2}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 2 } = \color{blue}{ 2 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{1}&11&-9&0&-17&27&-13&39\\& & 11& 2& \color{blue}{2} & & & \\ \hline &11&2&\color{blue}{2}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ 2 } = \color{orangered}{ -15 } $
$$ \begin{array}{c|rrrrrrr}1&11&-9&0&\color{orangered}{ -17 }&27&-13&39\\& & 11& 2& \color{orangered}{2} & & & \\ \hline &11&2&2&\color{orangered}{-15}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \left( -15 \right) } = \color{blue}{ -15 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{1}&11&-9&0&-17&27&-13&39\\& & 11& 2& 2& \color{blue}{-15} & & \\ \hline &11&2&2&\color{blue}{-15}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 27 } + \color{orangered}{ \left( -15 \right) } = \color{orangered}{ 12 } $
$$ \begin{array}{c|rrrrrrr}1&11&-9&0&-17&\color{orangered}{ 27 }&-13&39\\& & 11& 2& 2& \color{orangered}{-15} & & \\ \hline &11&2&2&-15&\color{orangered}{12}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 12 } = \color{blue}{ 12 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{1}&11&-9&0&-17&27&-13&39\\& & 11& 2& 2& -15& \color{blue}{12} & \\ \hline &11&2&2&-15&\color{blue}{12}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -13 } + \color{orangered}{ 12 } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrrrrr}1&11&-9&0&-17&27&\color{orangered}{ -13 }&39\\& & 11& 2& 2& -15& \color{orangered}{12} & \\ \hline &11&2&2&-15&12&\color{orangered}{-1}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ -1 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{1}&11&-9&0&-17&27&-13&39\\& & 11& 2& 2& -15& 12& \color{blue}{-1} \\ \hline &11&2&2&-15&12&\color{blue}{-1}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 39 } + \color{orangered}{ \left( -1 \right) } = \color{orangered}{ 38 } $
$$ \begin{array}{c|rrrrrrr}1&11&-9&0&-17&27&-13&\color{orangered}{ 39 }\\& & 11& 2& 2& -15& 12& \color{orangered}{-1} \\ \hline &\color{blue}{11}&\color{blue}{2}&\color{blue}{2}&\color{blue}{-15}&\color{blue}{12}&\color{blue}{-1}&\color{orangered}{38} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 11x^{5}+2x^{4}+2x^{3}-15x^{2}+12x-1 } $ with a remainder of $ \color{red}{ 38 } $.