Find remainder when $ 12x^{3}+11x^{2}+7x-17 $ is divided by $ 3x+2 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-2&12&11&7&-17\\& & -24& 26& \color{black}{-66} \\ \hline &\color{blue}{12}&\color{blue}{-13}&\color{blue}{33}&\color{orangered}{-83} \end{array} $$The remainder when $ 12x^{3}+11x^{2}+7x-17 $ is divided by $ x+2 $ is $ \, \color{red}{ -83 } $.
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.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&12&11&7&-17\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-2&\color{orangered}{ 12 }&11&7&-17\\& & & & \\ \hline &\color{orangered}{12}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 12 } = \color{blue}{ -24 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&12&11&7&-17\\& & \color{blue}{-24} & & \\ \hline &\color{blue}{12}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 11 } + \color{orangered}{ \left( -24 \right) } = \color{orangered}{ -13 } $
$$ \begin{array}{c|rrrr}-2&12&\color{orangered}{ 11 }&7&-17\\& & \color{orangered}{-24} & & \\ \hline &12&\color{orangered}{-13}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -13 \right) } = \color{blue}{ 26 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&12&11&7&-17\\& & -24& \color{blue}{26} & \\ \hline &12&\color{blue}{-13}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 7 } + \color{orangered}{ 26 } = \color{orangered}{ 33 } $
$$ \begin{array}{c|rrrr}-2&12&11&\color{orangered}{ 7 }&-17\\& & -24& \color{orangered}{26} & \\ \hline &12&-13&\color{orangered}{33}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 33 } = \color{blue}{ -66 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&12&11&7&-17\\& & -24& 26& \color{blue}{-66} \\ \hline &12&-13&\color{blue}{33}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ \left( -66 \right) } = \color{orangered}{ -83 } $
$$ \begin{array}{c|rrrr}-2&12&11&7&\color{orangered}{ -17 }\\& & -24& 26& \color{orangered}{-66} \\ \hline &\color{blue}{12}&\color{blue}{-13}&\color{blue}{33}&\color{orangered}{-83} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -83 }\right) $.