Find remainder when $ 3x^{4}-16x^{3}-7x^{2}+64x-20 $ is divided by $ x+20 $.
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-20&3&-16&-7&64&-20\\& & -60& 1520& -30260& \color{black}{603920} \\ \hline &\color{blue}{3}&\color{blue}{-76}&\color{blue}{1513}&\color{blue}{-30196}&\color{orangered}{603900} \end{array} $$The remainder when $ 3x^{4}-16x^{3}-7x^{2}+64x-20 $ is divided by $ x+20 $ is $ \, \color{red}{ 603900 } $.
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 + 20 = 0 $ ( $ x = \color{blue}{ -20 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-20}&3&-16&-7&64&-20\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-20&\color{orangered}{ 3 }&-16&-7&64&-20\\& & & & & \\ \hline &\color{orangered}{3}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -20 } \cdot \color{blue}{ 3 } = \color{blue}{ -60 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-20}&3&-16&-7&64&-20\\& & \color{blue}{-60} & & & \\ \hline &\color{blue}{3}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -16 } + \color{orangered}{ \left( -60 \right) } = \color{orangered}{ -76 } $
$$ \begin{array}{c|rrrrr}-20&3&\color{orangered}{ -16 }&-7&64&-20\\& & \color{orangered}{-60} & & & \\ \hline &3&\color{orangered}{-76}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -20 } \cdot \color{blue}{ \left( -76 \right) } = \color{blue}{ 1520 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-20}&3&-16&-7&64&-20\\& & -60& \color{blue}{1520} & & \\ \hline &3&\color{blue}{-76}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ 1520 } = \color{orangered}{ 1513 } $
$$ \begin{array}{c|rrrrr}-20&3&-16&\color{orangered}{ -7 }&64&-20\\& & -60& \color{orangered}{1520} & & \\ \hline &3&-76&\color{orangered}{1513}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -20 } \cdot \color{blue}{ 1513 } = \color{blue}{ -30260 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-20}&3&-16&-7&64&-20\\& & -60& 1520& \color{blue}{-30260} & \\ \hline &3&-76&\color{blue}{1513}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 64 } + \color{orangered}{ \left( -30260 \right) } = \color{orangered}{ -30196 } $
$$ \begin{array}{c|rrrrr}-20&3&-16&-7&\color{orangered}{ 64 }&-20\\& & -60& 1520& \color{orangered}{-30260} & \\ \hline &3&-76&1513&\color{orangered}{-30196}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -20 } \cdot \color{blue}{ \left( -30196 \right) } = \color{blue}{ 603920 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-20}&3&-16&-7&64&-20\\& & -60& 1520& -30260& \color{blue}{603920} \\ \hline &3&-76&1513&\color{blue}{-30196}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -20 } + \color{orangered}{ 603920 } = \color{orangered}{ 603900 } $
$$ \begin{array}{c|rrrrr}-20&3&-16&-7&64&\color{orangered}{ -20 }\\& & -60& 1520& -30260& \color{orangered}{603920} \\ \hline &\color{blue}{3}&\color{blue}{-76}&\color{blue}{1513}&\color{blue}{-30196}&\color{orangered}{603900} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 603900 }\right) $.