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

The synthetic division table is:

$$ \begin{array}{c|rrr}-3&-3&4&4\\& & 9& \color{black}{-39} \\ \hline &\color{blue}{-3}&\color{blue}{13}&\color{orangered}{-35} \end{array} $$

The remainder when $ -3x^{2}+4x+4 $ is divided by $ x+3 $ is $ \, \color{red}{ -35 } $.

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 + 3 = 0 $ ( $ x = \color{blue}{ -3 } $ ) at the left.

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

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

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

Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ 9 } $.

$$ \begin{array}{c|rrr}\color{blue}{-3}&-3&4&4\\& & \color{blue}{9} & \\ \hline &\color{blue}{-3}&& \end{array} $$

Step 3 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 9 } = \color{orangered}{ 13 } $

$$ \begin{array}{c|rrr}-3&-3&\color{orangered}{ 4 }&4\\& & \color{orangered}{9} & \\ \hline &-3&\color{orangered}{13}& \end{array} $$

Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 13 } = \color{blue}{ -39 } $.

$$ \begin{array}{c|rrr}\color{blue}{-3}&-3&4&4\\& & 9& \color{blue}{-39} \\ \hline &-3&\color{blue}{13}& \end{array} $$

Step 5 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ \left( -39 \right) } = \color{orangered}{ -35 } $

$$ \begin{array}{c|rrr}-3&-3&4&\color{orangered}{ 4 }\\& & 9& \color{orangered}{-39} \\ \hline &\color{blue}{-3}&\color{blue}{13}&\color{orangered}{-35} \end{array} $$

Remainder is the last entry in the bottom row $ \left(\color{red}{ -35 }\right) $.

Synthetic Division Calculator