Find remainder when $ -6x^{2}+2x+1 $ is divided by $ x+6 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-6&-6&2&1\\& & 36& \color{black}{-228} \\ \hline &\color{blue}{-6}&\color{blue}{38}&\color{orangered}{-227} \end{array} $$The remainder when $ -6x^{2}+2x+1 $ is divided by $ x+6 $ is $ \, \color{red}{ -227 } $.
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.
$$ \begin{array}{c|rrr}\color{blue}{-6}&-6&2&1\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-6&\color{orangered}{ -6 }&2&1\\& & & \\ \hline &\color{orangered}{-6}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ 36 } $.
$$ \begin{array}{c|rrr}\color{blue}{-6}&-6&2&1\\& & \color{blue}{36} & \\ \hline &\color{blue}{-6}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 36 } = \color{orangered}{ 38 } $
$$ \begin{array}{c|rrr}-6&-6&\color{orangered}{ 2 }&1\\& & \color{orangered}{36} & \\ \hline &-6&\color{orangered}{38}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 38 } = \color{blue}{ -228 } $.
$$ \begin{array}{c|rrr}\color{blue}{-6}&-6&2&1\\& & 36& \color{blue}{-228} \\ \hline &-6&\color{blue}{38}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ \left( -228 \right) } = \color{orangered}{ -227 } $
$$ \begin{array}{c|rrr}-6&-6&2&\color{orangered}{ 1 }\\& & 36& \color{orangered}{-228} \\ \hline &\color{blue}{-6}&\color{blue}{38}&\color{orangered}{-227} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -227 }\right) $.