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