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