Find remainder when $ x^{2}-3x-33 $ is divided by $ x+4 $.

The synthetic division table is:

$$ \begin{array}{c|rrr}-4&1&-3&-33\\& & -4& \color{black}{28} \\ \hline &\color{blue}{1}&\color{blue}{-7}&\color{orangered}{-5} \end{array} $$

The remainder when $ x^{2}-3x-33 $ is divided by $ x+4 $ is $ \, \color{red}{ -5 } $.

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.

$$ \begin{array}{c|rrr}\color{blue}{-4}&1&-3&-33\\& & & \\ \hline &&& \end{array} $$

Step 1 : Bring down the leading coefficient to the bottom row.

$$ \begin{array}{c|rrr}-4&\color{orangered}{ 1 }&-3&-33\\& & & \\ \hline &\color{orangered}{1}&& \end{array} $$

Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 1 } = \color{blue}{ -4 } $.

$$ \begin{array}{c|rrr}\color{blue}{-4}&1&-3&-33\\& & \color{blue}{-4} & \\ \hline &\color{blue}{1}&& \end{array} $$

Step 3 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ -7 } $

$$ \begin{array}{c|rrr}-4&1&\color{orangered}{ -3 }&-33\\& & \color{orangered}{-4} & \\ \hline &1&\color{orangered}{-7}& \end{array} $$

Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -7 \right) } = \color{blue}{ 28 } $.

$$ \begin{array}{c|rrr}\color{blue}{-4}&1&-3&-33\\& & -4& \color{blue}{28} \\ \hline &1&\color{blue}{-7}& \end{array} $$

Step 5 : Add down last column: $ \color{orangered}{ -33 } + \color{orangered}{ 28 } = \color{orangered}{ -5 } $

$$ \begin{array}{c|rrr}-4&1&-3&\color{orangered}{ -33 }\\& & -4& \color{orangered}{28} \\ \hline &\color{blue}{1}&\color{blue}{-7}&\color{orangered}{-5} \end{array} $$

Remainder is the last entry in the bottom row $ \left(\color{red}{ -5 }\right) $.

Synthetic Division Calculator