Find remainder when $ x^{3}-27x^{2}+45 $ is divided by $ x+6 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-6&1&-27&0&45\\& & -6& 198& \color{black}{-1188} \\ \hline &\color{blue}{1}&\color{blue}{-33}&\color{blue}{198}&\color{orangered}{-1143} \end{array} $$The remainder when $ x^{3}-27x^{2}+45 $ is divided by $ x+6 $ is $ \, \color{red}{ -1143 } $.
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.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&-27&0&45\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-6&\color{orangered}{ 1 }&-27&0&45\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&-27&0&45\\& & \color{blue}{-6} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -27 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -33 } $
$$ \begin{array}{c|rrrr}-6&1&\color{orangered}{ -27 }&0&45\\& & \color{orangered}{-6} & & \\ \hline &1&\color{orangered}{-33}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -33 \right) } = \color{blue}{ 198 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&-27&0&45\\& & -6& \color{blue}{198} & \\ \hline &1&\color{blue}{-33}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 198 } = \color{orangered}{ 198 } $
$$ \begin{array}{c|rrrr}-6&1&-27&\color{orangered}{ 0 }&45\\& & -6& \color{orangered}{198} & \\ \hline &1&-33&\color{orangered}{198}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 198 } = \color{blue}{ -1188 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&-27&0&45\\& & -6& 198& \color{blue}{-1188} \\ \hline &1&-33&\color{blue}{198}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 45 } + \color{orangered}{ \left( -1188 \right) } = \color{orangered}{ -1143 } $
$$ \begin{array}{c|rrrr}-6&1&-27&0&\color{orangered}{ 45 }\\& & -6& 198& \color{orangered}{-1188} \\ \hline &\color{blue}{1}&\color{blue}{-33}&\color{blue}{198}&\color{orangered}{-1143} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -1143 }\right) $.