Find remainder when $ 18x^{5}+30x^{4}-24x^{3}-17x^{2}+15x+8 $ is divided by $ x+2 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-2&18&30&-24&-17&15&8\\& & -36& 12& 24& -14& \color{black}{-2} \\ \hline &\color{blue}{18}&\color{blue}{-6}&\color{blue}{-12}&\color{blue}{7}&\color{blue}{1}&\color{orangered}{6} \end{array} $$The remainder when $ 18x^{5}+30x^{4}-24x^{3}-17x^{2}+15x+8 $ is divided by $ x+2 $ is $ \, \color{red}{ 6 } $.
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 + 2 = 0 $ ( $ x = \color{blue}{ -2 } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&18&30&-24&-17&15&8\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-2&\color{orangered}{ 18 }&30&-24&-17&15&8\\& & & & & & \\ \hline &\color{orangered}{18}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 18 } = \color{blue}{ -36 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&18&30&-24&-17&15&8\\& & \color{blue}{-36} & & & & \\ \hline &\color{blue}{18}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 30 } + \color{orangered}{ \left( -36 \right) } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrrrr}-2&18&\color{orangered}{ 30 }&-24&-17&15&8\\& & \color{orangered}{-36} & & & & \\ \hline &18&\color{orangered}{-6}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ 12 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&18&30&-24&-17&15&8\\& & -36& \color{blue}{12} & & & \\ \hline &18&\color{blue}{-6}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -24 } + \color{orangered}{ 12 } = \color{orangered}{ -12 } $
$$ \begin{array}{c|rrrrrr}-2&18&30&\color{orangered}{ -24 }&-17&15&8\\& & -36& \color{orangered}{12} & & & \\ \hline &18&-6&\color{orangered}{-12}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -12 \right) } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&18&30&-24&-17&15&8\\& & -36& 12& \color{blue}{24} & & \\ \hline &18&-6&\color{blue}{-12}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ 24 } = \color{orangered}{ 7 } $
$$ \begin{array}{c|rrrrrr}-2&18&30&-24&\color{orangered}{ -17 }&15&8\\& & -36& 12& \color{orangered}{24} & & \\ \hline &18&-6&-12&\color{orangered}{7}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 7 } = \color{blue}{ -14 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&18&30&-24&-17&15&8\\& & -36& 12& 24& \color{blue}{-14} & \\ \hline &18&-6&-12&\color{blue}{7}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 15 } + \color{orangered}{ \left( -14 \right) } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrrrrr}-2&18&30&-24&-17&\color{orangered}{ 15 }&8\\& & -36& 12& 24& \color{orangered}{-14} & \\ \hline &18&-6&-12&7&\color{orangered}{1}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 1 } = \color{blue}{ -2 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&18&30&-24&-17&15&8\\& & -36& 12& 24& -14& \color{blue}{-2} \\ \hline &18&-6&-12&7&\color{blue}{1}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrrrr}-2&18&30&-24&-17&15&\color{orangered}{ 8 }\\& & -36& 12& 24& -14& \color{orangered}{-2} \\ \hline &\color{blue}{18}&\color{blue}{-6}&\color{blue}{-12}&\color{blue}{7}&\color{blue}{1}&\color{orangered}{6} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 6 }\right) $.