Find remainder when $ 3x^{4}-16x^{3}-7x^{2}+64x-20 $ is divided by $ x+10 $.
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-10&3&-16&-7&64&-20\\& & -30& 460& -4530& \color{black}{44660} \\ \hline &\color{blue}{3}&\color{blue}{-46}&\color{blue}{453}&\color{blue}{-4466}&\color{orangered}{44640} \end{array} $$The remainder when $ 3x^{4}-16x^{3}-7x^{2}+64x-20 $ is divided by $ x+10 $ is $ \, \color{red}{ 44640 } $.
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 + 10 = 0 $ ( $ x = \color{blue}{ -10 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-10}&3&-16&-7&64&-20\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-10&\color{orangered}{ 3 }&-16&-7&64&-20\\& & & & & \\ \hline &\color{orangered}{3}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -10 } \cdot \color{blue}{ 3 } = \color{blue}{ -30 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-10}&3&-16&-7&64&-20\\& & \color{blue}{-30} & & & \\ \hline &\color{blue}{3}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -16 } + \color{orangered}{ \left( -30 \right) } = \color{orangered}{ -46 } $
$$ \begin{array}{c|rrrrr}-10&3&\color{orangered}{ -16 }&-7&64&-20\\& & \color{orangered}{-30} & & & \\ \hline &3&\color{orangered}{-46}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -10 } \cdot \color{blue}{ \left( -46 \right) } = \color{blue}{ 460 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-10}&3&-16&-7&64&-20\\& & -30& \color{blue}{460} & & \\ \hline &3&\color{blue}{-46}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ 460 } = \color{orangered}{ 453 } $
$$ \begin{array}{c|rrrrr}-10&3&-16&\color{orangered}{ -7 }&64&-20\\& & -30& \color{orangered}{460} & & \\ \hline &3&-46&\color{orangered}{453}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -10 } \cdot \color{blue}{ 453 } = \color{blue}{ -4530 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-10}&3&-16&-7&64&-20\\& & -30& 460& \color{blue}{-4530} & \\ \hline &3&-46&\color{blue}{453}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 64 } + \color{orangered}{ \left( -4530 \right) } = \color{orangered}{ -4466 } $
$$ \begin{array}{c|rrrrr}-10&3&-16&-7&\color{orangered}{ 64 }&-20\\& & -30& 460& \color{orangered}{-4530} & \\ \hline &3&-46&453&\color{orangered}{-4466}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -10 } \cdot \color{blue}{ \left( -4466 \right) } = \color{blue}{ 44660 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-10}&3&-16&-7&64&-20\\& & -30& 460& -4530& \color{blue}{44660} \\ \hline &3&-46&453&\color{blue}{-4466}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -20 } + \color{orangered}{ 44660 } = \color{orangered}{ 44640 } $
$$ \begin{array}{c|rrrrr}-10&3&-16&-7&64&\color{orangered}{ -20 }\\& & -30& 460& -4530& \color{orangered}{44660} \\ \hline &\color{blue}{3}&\color{blue}{-46}&\color{blue}{453}&\color{blue}{-4466}&\color{orangered}{44640} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 44640 }\right) $.