Find remainder when $ 6x^{4}+30x^{3}+26x^{2}+12x+21 $ is divided by $ x+4 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-4&6&30&26&12&21\\& & -24& -24& -8& \color{black}{-16} \\ \hline &\color{blue}{6}&\color{blue}{6}&\color{blue}{2}&\color{blue}{4}&\color{orangered}{5} \end{array} $$The remainder when $ 6x^{4}+30x^{3}+26x^{2}+12x+21 $ is divided by $ x+4 $ is $ \, \color{red}{ 5 } $.
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 + 4 = 0 $ ( $ x = \color{blue}{ -4 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&6&30&26&12&21\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-4&\color{orangered}{ 6 }&30&26&12&21\\& & & & & \\ \hline &\color{orangered}{6}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 6 } = \color{blue}{ -24 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&6&30&26&12&21\\& & \color{blue}{-24} & & & \\ \hline &\color{blue}{6}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 30 } + \color{orangered}{ \left( -24 \right) } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrrr}-4&6&\color{orangered}{ 30 }&26&12&21\\& & \color{orangered}{-24} & & & \\ \hline &6&\color{orangered}{6}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 6 } = \color{blue}{ -24 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&6&30&26&12&21\\& & -24& \color{blue}{-24} & & \\ \hline &6&\color{blue}{6}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 26 } + \color{orangered}{ \left( -24 \right) } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrrr}-4&6&30&\color{orangered}{ 26 }&12&21\\& & -24& \color{orangered}{-24} & & \\ \hline &6&6&\color{orangered}{2}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 2 } = \color{blue}{ -8 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&6&30&26&12&21\\& & -24& -24& \color{blue}{-8} & \\ \hline &6&6&\color{blue}{2}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 12 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrrr}-4&6&30&26&\color{orangered}{ 12 }&21\\& & -24& -24& \color{orangered}{-8} & \\ \hline &6&6&2&\color{orangered}{4}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 4 } = \color{blue}{ -16 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&6&30&26&12&21\\& & -24& -24& -8& \color{blue}{-16} \\ \hline &6&6&2&\color{blue}{4}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 21 } + \color{orangered}{ \left( -16 \right) } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrrrr}-4&6&30&26&12&\color{orangered}{ 21 }\\& & -24& -24& -8& \color{orangered}{-16} \\ \hline &\color{blue}{6}&\color{blue}{6}&\color{blue}{2}&\color{blue}{4}&\color{orangered}{5} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 5 }\right) $.