Find remainder when $ 3x^{5}+11x^{4}-19x^{3}+9x^{2}+19x-36 $ is divided by $ x+5 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-5&3&11&-19&9&19&-36\\& & -15& 20& -5& -20& \color{black}{5} \\ \hline &\color{blue}{3}&\color{blue}{-4}&\color{blue}{1}&\color{blue}{4}&\color{blue}{-1}&\color{orangered}{-31} \end{array} $$The remainder when $ 3x^{5}+11x^{4}-19x^{3}+9x^{2}+19x-36 $ is divided by $ x+5 $ is $ \, \color{red}{ -31 } $.
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 + 5 = 0 $ ( $ x = \color{blue}{ -5 } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{-5}&3&11&-19&9&19&-36\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-5&\color{orangered}{ 3 }&11&-19&9&19&-36\\& & & & & & \\ \hline &\color{orangered}{3}&&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-5}&3&11&-19&9&19&-36\\& & \color{blue}{-15} & & & & \\ \hline &\color{blue}{3}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 11 } + \color{orangered}{ \left( -15 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrrrr}-5&3&\color{orangered}{ 11 }&-19&9&19&-36\\& & \color{orangered}{-15} & & & & \\ \hline &3&\color{orangered}{-4}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 20 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-5}&3&11&-19&9&19&-36\\& & -15& \color{blue}{20} & & & \\ \hline &3&\color{blue}{-4}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -19 } + \color{orangered}{ 20 } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrrrrr}-5&3&11&\color{orangered}{ -19 }&9&19&-36\\& & -15& \color{orangered}{20} & & & \\ \hline &3&-4&\color{orangered}{1}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 1 } = \color{blue}{ -5 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-5}&3&11&-19&9&19&-36\\& & -15& 20& \color{blue}{-5} & & \\ \hline &3&-4&\color{blue}{1}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ \left( -5 \right) } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrrrr}-5&3&11&-19&\color{orangered}{ 9 }&19&-36\\& & -15& 20& \color{orangered}{-5} & & \\ \hline &3&-4&1&\color{orangered}{4}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 4 } = \color{blue}{ -20 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-5}&3&11&-19&9&19&-36\\& & -15& 20& -5& \color{blue}{-20} & \\ \hline &3&-4&1&\color{blue}{4}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 19 } + \color{orangered}{ \left( -20 \right) } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrrrr}-5&3&11&-19&9&\color{orangered}{ 19 }&-36\\& & -15& 20& -5& \color{orangered}{-20} & \\ \hline &3&-4&1&4&\color{orangered}{-1}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 5 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-5}&3&11&-19&9&19&-36\\& & -15& 20& -5& -20& \color{blue}{5} \\ \hline &3&-4&1&4&\color{blue}{-1}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -36 } + \color{orangered}{ 5 } = \color{orangered}{ -31 } $
$$ \begin{array}{c|rrrrrr}-5&3&11&-19&9&19&\color{orangered}{ -36 }\\& & -15& 20& -5& -20& \color{orangered}{5} \\ \hline &\color{blue}{3}&\color{blue}{-4}&\color{blue}{1}&\color{blue}{4}&\color{blue}{-1}&\color{orangered}{-31} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -31 }\right) $.