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