Find remainder when $ 3x^{4}-16x^{3}-7x^{2}+64x-20 $ is divided by $ x $.
The synthetic division table is:
$$ \begin{array}{c|rrrrr}0&3&-16&-7&64&-20\\& & 0& 0& 0& \color{black}{0} \\ \hline &\color{blue}{3}&\color{blue}{-16}&\color{blue}{-7}&\color{blue}{64}&\color{orangered}{-20} \end{array} $$The remainder when $ 3x^{4}-16x^{3}-7x^{2}+64x-20 $ is divided by $ x $ is $ \, \color{red}{ -20 } $.
We can find remainder using synthetic division method.
Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&3&-16&-7&64&-20\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}0&\color{orangered}{ 3 }&-16&-7&64&-20\\& & & & & \\ \hline &\color{orangered}{3}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 3 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&3&-16&-7&64&-20\\& & \color{blue}{0} & & & \\ \hline &\color{blue}{3}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -16 } + \color{orangered}{ 0 } = \color{orangered}{ -16 } $
$$ \begin{array}{c|rrrrr}0&3&\color{orangered}{ -16 }&-7&64&-20\\& & \color{orangered}{0} & & & \\ \hline &3&\color{orangered}{-16}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -16 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&3&-16&-7&64&-20\\& & 0& \color{blue}{0} & & \\ \hline &3&\color{blue}{-16}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ 0 } = \color{orangered}{ -7 } $
$$ \begin{array}{c|rrrrr}0&3&-16&\color{orangered}{ -7 }&64&-20\\& & 0& \color{orangered}{0} & & \\ \hline &3&-16&\color{orangered}{-7}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -7 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&3&-16&-7&64&-20\\& & 0& 0& \color{blue}{0} & \\ \hline &3&-16&\color{blue}{-7}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 64 } + \color{orangered}{ 0 } = \color{orangered}{ 64 } $
$$ \begin{array}{c|rrrrr}0&3&-16&-7&\color{orangered}{ 64 }&-20\\& & 0& 0& \color{orangered}{0} & \\ \hline &3&-16&-7&\color{orangered}{64}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 64 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&3&-16&-7&64&-20\\& & 0& 0& 0& \color{blue}{0} \\ \hline &3&-16&-7&\color{blue}{64}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -20 } + \color{orangered}{ 0 } = \color{orangered}{ -20 } $
$$ \begin{array}{c|rrrrr}0&3&-16&-7&64&\color{orangered}{ -20 }\\& & 0& 0& 0& \color{orangered}{0} \\ \hline &\color{blue}{3}&\color{blue}{-16}&\color{blue}{-7}&\color{blue}{64}&\color{orangered}{-20} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -20 }\right) $.