Find remainder when $ 5x^{6}-80x^{4}+2x^{2}-32 $ is divided by $ x+4 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}-4&5&0&-80&0&2&0&-32\\& & -20& 80& 0& 0& -8& \color{black}{32} \\ \hline &\color{blue}{5}&\color{blue}{-20}&\color{blue}{0}&\color{blue}{0}&\color{blue}{2}&\color{blue}{-8}&\color{orangered}{0} \end{array} $$The remainder when $ 5x^{6}-80x^{4}+2x^{2}-32 $ is divided by $ x+4 $ is $ \, \color{red}{ 0 } $.
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.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-4}&5&0&-80&0&2&0&-32\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}-4&\color{orangered}{ 5 }&0&-80&0&2&0&-32\\& & & & & & & \\ \hline &\color{orangered}{5}&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 5 } = \color{blue}{ -20 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-4}&5&0&-80&0&2&0&-32\\& & \color{blue}{-20} & & & & & \\ \hline &\color{blue}{5}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -20 \right) } = \color{orangered}{ -20 } $
$$ \begin{array}{c|rrrrrrr}-4&5&\color{orangered}{ 0 }&-80&0&2&0&-32\\& & \color{orangered}{-20} & & & & & \\ \hline &5&\color{orangered}{-20}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -20 \right) } = \color{blue}{ 80 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-4}&5&0&-80&0&2&0&-32\\& & -20& \color{blue}{80} & & & & \\ \hline &5&\color{blue}{-20}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -80 } + \color{orangered}{ 80 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrrr}-4&5&0&\color{orangered}{ -80 }&0&2&0&-32\\& & -20& \color{orangered}{80} & & & & \\ \hline &5&-20&\color{orangered}{0}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-4}&5&0&-80&0&2&0&-32\\& & -20& 80& \color{blue}{0} & & & \\ \hline &5&-20&\color{blue}{0}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrrr}-4&5&0&-80&\color{orangered}{ 0 }&2&0&-32\\& & -20& 80& \color{orangered}{0} & & & \\ \hline &5&-20&0&\color{orangered}{0}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-4}&5&0&-80&0&2&0&-32\\& & -20& 80& 0& \color{blue}{0} & & \\ \hline &5&-20&0&\color{blue}{0}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 0 } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrrrrr}-4&5&0&-80&0&\color{orangered}{ 2 }&0&-32\\& & -20& 80& 0& \color{orangered}{0} & & \\ \hline &5&-20&0&0&\color{orangered}{2}&& \end{array} $$Step 10 : 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 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-4}&5&0&-80&0&2&0&-32\\& & -20& 80& 0& 0& \color{blue}{-8} & \\ \hline &5&-20&0&0&\color{blue}{2}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrrrrrr}-4&5&0&-80&0&2&\color{orangered}{ 0 }&-32\\& & -20& 80& 0& 0& \color{orangered}{-8} & \\ \hline &5&-20&0&0&2&\color{orangered}{-8}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ 32 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-4}&5&0&-80&0&2&0&-32\\& & -20& 80& 0& 0& -8& \color{blue}{32} \\ \hline &5&-20&0&0&2&\color{blue}{-8}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ -32 } + \color{orangered}{ 32 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrrr}-4&5&0&-80&0&2&0&\color{orangered}{ -32 }\\& & -20& 80& 0& 0& -8& \color{orangered}{32} \\ \hline &\color{blue}{5}&\color{blue}{-20}&\color{blue}{0}&\color{blue}{0}&\color{blue}{2}&\color{blue}{-8}&\color{orangered}{0} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 0 }\right) $.