Find remainder when $ 6x^{3}+29x^{2}+16x-16 $ is divided by $ x+4 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-4&6&29&16&-16\\& & -24& -20& \color{black}{16} \\ \hline &\color{blue}{6}&\color{blue}{5}&\color{blue}{-4}&\color{orangered}{0} \end{array} $$The remainder when $ 6x^{3}+29x^{2}+16x-16 $ is divided by $ x+4 $ is $ \, \color{red}{ 0 } $.
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 + 4 = 0 $ ( $ x = \color{blue}{ -4 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&6&29&16&-16\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-4&\color{orangered}{ 6 }&29&16&-16\\& & & & \\ \hline &\color{orangered}{6}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 6 } = \color{blue}{ -24 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&6&29&16&-16\\& & \color{blue}{-24} & & \\ \hline &\color{blue}{6}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 29 } + \color{orangered}{ \left( -24 \right) } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrrr}-4&6&\color{orangered}{ 29 }&16&-16\\& & \color{orangered}{-24} & & \\ \hline &6&\color{orangered}{5}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 5 } = \color{blue}{ -20 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&6&29&16&-16\\& & -24& \color{blue}{-20} & \\ \hline &6&\color{blue}{5}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 16 } + \color{orangered}{ \left( -20 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrr}-4&6&29&\color{orangered}{ 16 }&-16\\& & -24& \color{orangered}{-20} & \\ \hline &6&5&\color{orangered}{-4}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 16 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&6&29&16&-16\\& & -24& -20& \color{blue}{16} \\ \hline &6&5&\color{blue}{-4}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -16 } + \color{orangered}{ 16 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}-4&6&29&16&\color{orangered}{ -16 }\\& & -24& -20& \color{orangered}{16} \\ \hline &\color{blue}{6}&\color{blue}{5}&\color{blue}{-4}&\color{orangered}{0} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 0 }\right) $.