Divide using synthetic division $$ ( 6x^{4}-41x^{3}+25x^{2}+29x+15 ) \div ( x-6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}6&6&-41&25&29&15\\& & 36& -30& -30& \color{black}{-6} \\ \hline &\color{blue}{6}&\color{blue}{-5}&\color{blue}{-5}&\color{blue}{-1}&\color{orangered}{9} \end{array} $$The solution is:
$$ \frac{ 6x^{4}-41x^{3}+25x^{2}+29x+15 }{ x-6 } = \color{blue}{6x^{3}-5x^{2}-5x-1} ~+~ \frac{ \color{red}{ 9 } }{ x-6 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -6 = 0 $ ( $ x = \color{blue}{ 6 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{6}&6&-41&25&29&15\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}6&\color{orangered}{ 6 }&-41&25&29&15\\& & & & & \\ \hline &\color{orangered}{6}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 6 } = \color{blue}{ 36 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{6}&6&-41&25&29&15\\& & \color{blue}{36} & & & \\ \hline &\color{blue}{6}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -41 } + \color{orangered}{ 36 } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrrrr}6&6&\color{orangered}{ -41 }&25&29&15\\& & \color{orangered}{36} & & & \\ \hline &6&\color{orangered}{-5}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ -30 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{6}&6&-41&25&29&15\\& & 36& \color{blue}{-30} & & \\ \hline &6&\color{blue}{-5}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 25 } + \color{orangered}{ \left( -30 \right) } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrrrr}6&6&-41&\color{orangered}{ 25 }&29&15\\& & 36& \color{orangered}{-30} & & \\ \hline &6&-5&\color{orangered}{-5}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ -30 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{6}&6&-41&25&29&15\\& & 36& -30& \color{blue}{-30} & \\ \hline &6&-5&\color{blue}{-5}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 29 } + \color{orangered}{ \left( -30 \right) } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrrr}6&6&-41&25&\color{orangered}{ 29 }&15\\& & 36& -30& \color{orangered}{-30} & \\ \hline &6&-5&-5&\color{orangered}{-1}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{6}&6&-41&25&29&15\\& & 36& -30& -30& \color{blue}{-6} \\ \hline &6&-5&-5&\color{blue}{-1}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 15 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ 9 } $
$$ \begin{array}{c|rrrrr}6&6&-41&25&29&\color{orangered}{ 15 }\\& & 36& -30& -30& \color{orangered}{-6} \\ \hline &\color{blue}{6}&\color{blue}{-5}&\color{blue}{-5}&\color{blue}{-1}&\color{orangered}{9} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 6x^{3}-5x^{2}-5x-1 } $ with a remainder of $ \color{red}{ 9 } $.