Find remainder when $ x^{4}-3x^{3}+37x-68 $ is divided by $ x+6 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-6&1&-3&0&37&-68\\& & -6& 54& -324& \color{black}{1722} \\ \hline &\color{blue}{1}&\color{blue}{-9}&\color{blue}{54}&\color{blue}{-287}&\color{orangered}{1654} \end{array} $$The remainder when $ x^{4}-3x^{3}+37x-68 $ is divided by $ x+6 $ is $ \, \color{red}{ 1654 } $.
Explanation
We can find remainder using synthetic division method.
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.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&1&-3&0&37&-68\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-6&\color{orangered}{ 1 }&-3&0&37&-68\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&1&-3&0&37&-68\\& & \color{blue}{-6} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrrr}-6&1&\color{orangered}{ -3 }&0&37&-68\\& & \color{orangered}{-6} & & & \\ \hline &1&\color{orangered}{-9}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ 54 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&1&-3&0&37&-68\\& & -6& \color{blue}{54} & & \\ \hline &1&\color{blue}{-9}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 54 } = \color{orangered}{ 54 } $
$$ \begin{array}{c|rrrrr}-6&1&-3&\color{orangered}{ 0 }&37&-68\\& & -6& \color{orangered}{54} & & \\ \hline &1&-9&\color{orangered}{54}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 54 } = \color{blue}{ -324 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&1&-3&0&37&-68\\& & -6& 54& \color{blue}{-324} & \\ \hline &1&-9&\color{blue}{54}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 37 } + \color{orangered}{ \left( -324 \right) } = \color{orangered}{ -287 } $
$$ \begin{array}{c|rrrrr}-6&1&-3&0&\color{orangered}{ 37 }&-68\\& & -6& 54& \color{orangered}{-324} & \\ \hline &1&-9&54&\color{orangered}{-287}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -287 \right) } = \color{blue}{ 1722 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&1&-3&0&37&-68\\& & -6& 54& -324& \color{blue}{1722} \\ \hline &1&-9&54&\color{blue}{-287}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -68 } + \color{orangered}{ 1722 } = \color{orangered}{ 1654 } $
$$ \begin{array}{c|rrrrr}-6&1&-3&0&37&\color{orangered}{ -68 }\\& & -6& 54& -324& \color{orangered}{1722} \\ \hline &\color{blue}{1}&\color{blue}{-9}&\color{blue}{54}&\color{blue}{-287}&\color{orangered}{1654} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 1654 }\right) $.