Find remainder when $ x^{4}+3 $ is divided by $ x+4 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-4&1&0&0&0&3\\& & -4& 16& -64& \color{black}{256} \\ \hline &\color{blue}{1}&\color{blue}{-4}&\color{blue}{16}&\color{blue}{-64}&\color{orangered}{259} \end{array} $$The remainder when $ x^{4}+3 $ is divided by $ x+4 $ is $ \, \color{red}{ 259 } $.
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.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&1&0&0&0&3\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-4&\color{orangered}{ 1 }&0&0&0&3\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 1 } = \color{blue}{ -4 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&1&0&0&0&3\\& & \color{blue}{-4} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrrr}-4&1&\color{orangered}{ 0 }&0&0&3\\& & \color{orangered}{-4} & & & \\ \hline &1&\color{orangered}{-4}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 16 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&1&0&0&0&3\\& & -4& \color{blue}{16} & & \\ \hline &1&\color{blue}{-4}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 16 } = \color{orangered}{ 16 } $
$$ \begin{array}{c|rrrrr}-4&1&0&\color{orangered}{ 0 }&0&3\\& & -4& \color{orangered}{16} & & \\ \hline &1&-4&\color{orangered}{16}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 16 } = \color{blue}{ -64 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&1&0&0&0&3\\& & -4& 16& \color{blue}{-64} & \\ \hline &1&-4&\color{blue}{16}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -64 \right) } = \color{orangered}{ -64 } $
$$ \begin{array}{c|rrrrr}-4&1&0&0&\color{orangered}{ 0 }&3\\& & -4& 16& \color{orangered}{-64} & \\ \hline &1&-4&16&\color{orangered}{-64}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -64 \right) } = \color{blue}{ 256 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&1&0&0&0&3\\& & -4& 16& -64& \color{blue}{256} \\ \hline &1&-4&16&\color{blue}{-64}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 256 } = \color{orangered}{ 259 } $
$$ \begin{array}{c|rrrrr}-4&1&0&0&0&\color{orangered}{ 3 }\\& & -4& 16& -64& \color{orangered}{256} \\ \hline &\color{blue}{1}&\color{blue}{-4}&\color{blue}{16}&\color{blue}{-64}&\color{orangered}{259} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 259 }\right) $.