Find remainder when $ 16x^{3}+35x^{2}+12x+17 $ is divided by $ x+2 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-2&16&35&12&17\\& & -32& -6& \color{black}{-12} \\ \hline &\color{blue}{16}&\color{blue}{3}&\color{blue}{6}&\color{orangered}{5} \end{array} $$The remainder when $ 16x^{3}+35x^{2}+12x+17 $ is divided by $ x+2 $ is $ \, \color{red}{ 5 } $.
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.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&16&35&12&17\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-2&\color{orangered}{ 16 }&35&12&17\\& & & & \\ \hline &\color{orangered}{16}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 16 } = \color{blue}{ -32 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&16&35&12&17\\& & \color{blue}{-32} & & \\ \hline &\color{blue}{16}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 35 } + \color{orangered}{ \left( -32 \right) } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrr}-2&16&\color{orangered}{ 35 }&12&17\\& & \color{orangered}{-32} & & \\ \hline &16&\color{orangered}{3}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 3 } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&16&35&12&17\\& & -32& \color{blue}{-6} & \\ \hline &16&\color{blue}{3}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 12 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrr}-2&16&35&\color{orangered}{ 12 }&17\\& & -32& \color{orangered}{-6} & \\ \hline &16&3&\color{orangered}{6}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 6 } = \color{blue}{ -12 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&16&35&12&17\\& & -32& -6& \color{blue}{-12} \\ \hline &16&3&\color{blue}{6}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 17 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrrr}-2&16&35&12&\color{orangered}{ 17 }\\& & -32& -6& \color{orangered}{-12} \\ \hline &\color{blue}{16}&\color{blue}{3}&\color{blue}{6}&\color{orangered}{5} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 5 }\right) $.