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