Find remainder when $ 3x^{4}+20x^{3}+19x^{2}-33x-17 $ is divided by $ x+5 $.
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-5&3&20&19&-33&-17\\& & -15& -25& 30& \color{black}{15} \\ \hline &\color{blue}{3}&\color{blue}{5}&\color{blue}{-6}&\color{blue}{-3}&\color{orangered}{-2} \end{array} $$The remainder when $ 3x^{4}+20x^{3}+19x^{2}-33x-17 $ is divided by $ x+5 $ is $ \, \color{red}{ -2 } $.
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 + 5 = 0 $ ( $ x = \color{blue}{ -5 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&3&20&19&-33&-17\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-5&\color{orangered}{ 3 }&20&19&-33&-17\\& & & & & \\ \hline &\color{orangered}{3}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 3 } = \color{blue}{ -15 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&3&20&19&-33&-17\\& & \color{blue}{-15} & & & \\ \hline &\color{blue}{3}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 20 } + \color{orangered}{ \left( -15 \right) } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrrrr}-5&3&\color{orangered}{ 20 }&19&-33&-17\\& & \color{orangered}{-15} & & & \\ \hline &3&\color{orangered}{5}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 5 } = \color{blue}{ -25 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&3&20&19&-33&-17\\& & -15& \color{blue}{-25} & & \\ \hline &3&\color{blue}{5}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 19 } + \color{orangered}{ \left( -25 \right) } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrrr}-5&3&20&\color{orangered}{ 19 }&-33&-17\\& & -15& \color{orangered}{-25} & & \\ \hline &3&5&\color{orangered}{-6}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ 30 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&3&20&19&-33&-17\\& & -15& -25& \color{blue}{30} & \\ \hline &3&5&\color{blue}{-6}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -33 } + \color{orangered}{ 30 } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrrrr}-5&3&20&19&\color{orangered}{ -33 }&-17\\& & -15& -25& \color{orangered}{30} & \\ \hline &3&5&-6&\color{orangered}{-3}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ 15 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&3&20&19&-33&-17\\& & -15& -25& 30& \color{blue}{15} \\ \hline &3&5&-6&\color{blue}{-3}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ 15 } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrr}-5&3&20&19&-33&\color{orangered}{ -17 }\\& & -15& -25& 30& \color{orangered}{15} \\ \hline &\color{blue}{3}&\color{blue}{5}&\color{blue}{-6}&\color{blue}{-3}&\color{orangered}{-2} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -2 }\right) $.