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

The synthetic division table is:

$$ \begin{array}{c|rrr}4&1&-3&-33\\& & 4& \color{black}{4} \\ \hline &\color{blue}{1}&\color{blue}{1}&\color{orangered}{-29} \end{array} $$

The remainder when $ x^{2}-3x-33 $ is divided by $ x-4 $ is $ \, \color{red}{ -29 } $.

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}{ 4 } = \color{orangered}{ 1 } $

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

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

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

Step 5 : Add down last column: $ \color{orangered}{ -33 } + \color{orangered}{ 4 } = \color{orangered}{ -29 } $

$$ \begin{array}{c|rrr}4&1&-3&\color{orangered}{ -33 }\\& & 4& \color{orangered}{4} \\ \hline &\color{blue}{1}&\color{blue}{1}&\color{orangered}{-29} \end{array} $$

Remainder is the last entry in the bottom row $ \left(\color{red}{ -29 }\right) $.

Synthetic Division Calculator