Find remainder when $ x^{4}-18x^{3}+78x^{2}+294x-3355 $ is divided by $ -5x+6 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-6&1&-18&78&294&-3355\\& & -6& 144& -1332& \color{black}{6228} \\ \hline &\color{blue}{1}&\color{blue}{-24}&\color{blue}{222}&\color{blue}{-1038}&\color{orangered}{2873} \end{array} $$The remainder when $ x^{4}-18x^{3}+78x^{2}+294x-3355 $ is divided by $ x+6 $ is $ \, \color{red}{ 2873 } $.
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.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&1&-18&78&294&-3355\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-6&\color{orangered}{ 1 }&-18&78&294&-3355\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&1&-18&78&294&-3355\\& & \color{blue}{-6} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -18 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -24 } $
$$ \begin{array}{c|rrrrr}-6&1&\color{orangered}{ -18 }&78&294&-3355\\& & \color{orangered}{-6} & & & \\ \hline &1&\color{orangered}{-24}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -24 \right) } = \color{blue}{ 144 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&1&-18&78&294&-3355\\& & -6& \color{blue}{144} & & \\ \hline &1&\color{blue}{-24}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 78 } + \color{orangered}{ 144 } = \color{orangered}{ 222 } $
$$ \begin{array}{c|rrrrr}-6&1&-18&\color{orangered}{ 78 }&294&-3355\\& & -6& \color{orangered}{144} & & \\ \hline &1&-24&\color{orangered}{222}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 222 } = \color{blue}{ -1332 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&1&-18&78&294&-3355\\& & -6& 144& \color{blue}{-1332} & \\ \hline &1&-24&\color{blue}{222}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 294 } + \color{orangered}{ \left( -1332 \right) } = \color{orangered}{ -1038 } $
$$ \begin{array}{c|rrrrr}-6&1&-18&78&\color{orangered}{ 294 }&-3355\\& & -6& 144& \color{orangered}{-1332} & \\ \hline &1&-24&222&\color{orangered}{-1038}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -1038 \right) } = \color{blue}{ 6228 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&1&-18&78&294&-3355\\& & -6& 144& -1332& \color{blue}{6228} \\ \hline &1&-24&222&\color{blue}{-1038}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -3355 } + \color{orangered}{ 6228 } = \color{orangered}{ 2873 } $
$$ \begin{array}{c|rrrrr}-6&1&-18&78&294&\color{orangered}{ -3355 }\\& & -6& 144& -1332& \color{orangered}{6228} \\ \hline &\color{blue}{1}&\color{blue}{-24}&\color{blue}{222}&\color{blue}{-1038}&\color{orangered}{2873} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 2873 }\right) $.