Find remainder when $ 8x^{3}+44x^{2}+19x-10 $ is divided by $ x+5 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-5&8&44&19&-10\\& & -40& -20& \color{black}{5} \\ \hline &\color{blue}{8}&\color{blue}{4}&\color{blue}{-1}&\color{orangered}{-5} \end{array} $$The remainder when $ 8x^{3}+44x^{2}+19x-10 $ is divided by $ x+5 $ is $ \, \color{red}{ -5 } $.
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.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&8&44&19&-10\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-5&\color{orangered}{ 8 }&44&19&-10\\& & & & \\ \hline &\color{orangered}{8}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 8 } = \color{blue}{ -40 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&8&44&19&-10\\& & \color{blue}{-40} & & \\ \hline &\color{blue}{8}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 44 } + \color{orangered}{ \left( -40 \right) } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrr}-5&8&\color{orangered}{ 44 }&19&-10\\& & \color{orangered}{-40} & & \\ \hline &8&\color{orangered}{4}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 4 } = \color{blue}{ -20 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&8&44&19&-10\\& & -40& \color{blue}{-20} & \\ \hline &8&\color{blue}{4}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 19 } + \color{orangered}{ \left( -20 \right) } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrr}-5&8&44&\color{orangered}{ 19 }&-10\\& & -40& \color{orangered}{-20} & \\ \hline &8&4&\color{orangered}{-1}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 5 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&8&44&19&-10\\& & -40& -20& \color{blue}{5} \\ \hline &8&4&\color{blue}{-1}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ 5 } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrrr}-5&8&44&19&\color{orangered}{ -10 }\\& & -40& -20& \color{orangered}{5} \\ \hline &\color{blue}{8}&\color{blue}{4}&\color{blue}{-1}&\color{orangered}{-5} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -5 }\right) $.