Find remainder when $ 7x^{3}-29x+33 $ is divided by $ x+2 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-2&7&0&-29&33\\& & -14& 28& \color{black}{2} \\ \hline &\color{blue}{7}&\color{blue}{-14}&\color{blue}{-1}&\color{orangered}{35} \end{array} $$The remainder when $ 7x^{3}-29x+33 $ is divided by $ x+2 $ is $ \, \color{red}{ 35 } $.
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.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&7&0&-29&33\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-2&\color{orangered}{ 7 }&0&-29&33\\& & & & \\ \hline &\color{orangered}{7}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 7 } = \color{blue}{ -14 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&7&0&-29&33\\& & \color{blue}{-14} & & \\ \hline &\color{blue}{7}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -14 \right) } = \color{orangered}{ -14 } $
$$ \begin{array}{c|rrrr}-2&7&\color{orangered}{ 0 }&-29&33\\& & \color{orangered}{-14} & & \\ \hline &7&\color{orangered}{-14}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -14 \right) } = \color{blue}{ 28 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&7&0&-29&33\\& & -14& \color{blue}{28} & \\ \hline &7&\color{blue}{-14}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -29 } + \color{orangered}{ 28 } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrr}-2&7&0&\color{orangered}{ -29 }&33\\& & -14& \color{orangered}{28} & \\ \hline &7&-14&\color{orangered}{-1}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 2 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&7&0&-29&33\\& & -14& 28& \color{blue}{2} \\ \hline &7&-14&\color{blue}{-1}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 33 } + \color{orangered}{ 2 } = \color{orangered}{ 35 } $
$$ \begin{array}{c|rrrr}-2&7&0&-29&\color{orangered}{ 33 }\\& & -14& 28& \color{orangered}{2} \\ \hline &\color{blue}{7}&\color{blue}{-14}&\color{blue}{-1}&\color{orangered}{35} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 35 }\right) $.