Divide using synthetic division $$ ( x^{5}+6x^{4}-2x^{2}+6x+31 ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-3&1&6&0&-2&6&31\\& & -3& -9& 27& -75& \color{black}{207} \\ \hline &\color{blue}{1}&\color{blue}{3}&\color{blue}{-9}&\color{blue}{25}&\color{blue}{-69}&\color{orangered}{238} \end{array} $$The solution is:
$$ \frac{ x^{5}+6x^{4}-2x^{2}+6x+31 }{ x+3 } = \color{blue}{x^{4}+3x^{3}-9x^{2}+25x-69} ~+~ \frac{ \color{red}{ 238 } }{ x+3 } $$Explanation
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.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&1&6&0&-2&6&31\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-3&\color{orangered}{ 1 }&6&0&-2&6&31\\& & & & & & \\ \hline &\color{orangered}{1}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 1 } = \color{blue}{ -3 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&1&6&0&-2&6&31\\& & \color{blue}{-3} & & & & \\ \hline &\color{blue}{1}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ \left( -3 \right) } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrrrr}-3&1&\color{orangered}{ 6 }&0&-2&6&31\\& & \color{orangered}{-3} & & & & \\ \hline &1&\color{orangered}{3}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 3 } = \color{blue}{ -9 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&1&6&0&-2&6&31\\& & -3& \color{blue}{-9} & & & \\ \hline &1&\color{blue}{3}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -9 \right) } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrrrr}-3&1&6&\color{orangered}{ 0 }&-2&6&31\\& & -3& \color{orangered}{-9} & & & \\ \hline &1&3&\color{orangered}{-9}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ 27 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&1&6&0&-2&6&31\\& & -3& -9& \color{blue}{27} & & \\ \hline &1&3&\color{blue}{-9}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 27 } = \color{orangered}{ 25 } $
$$ \begin{array}{c|rrrrrr}-3&1&6&0&\color{orangered}{ -2 }&6&31\\& & -3& -9& \color{orangered}{27} & & \\ \hline &1&3&-9&\color{orangered}{25}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 25 } = \color{blue}{ -75 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&1&6&0&-2&6&31\\& & -3& -9& 27& \color{blue}{-75} & \\ \hline &1&3&-9&\color{blue}{25}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ \left( -75 \right) } = \color{orangered}{ -69 } $
$$ \begin{array}{c|rrrrrr}-3&1&6&0&-2&\color{orangered}{ 6 }&31\\& & -3& -9& 27& \color{orangered}{-75} & \\ \hline &1&3&-9&25&\color{orangered}{-69}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -69 \right) } = \color{blue}{ 207 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&1&6&0&-2&6&31\\& & -3& -9& 27& -75& \color{blue}{207} \\ \hline &1&3&-9&25&\color{blue}{-69}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 31 } + \color{orangered}{ 207 } = \color{orangered}{ 238 } $
$$ \begin{array}{c|rrrrrr}-3&1&6&0&-2&6&\color{orangered}{ 31 }\\& & -3& -9& 27& -75& \color{orangered}{207} \\ \hline &\color{blue}{1}&\color{blue}{3}&\color{blue}{-9}&\color{blue}{25}&\color{blue}{-69}&\color{orangered}{238} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{4}+3x^{3}-9x^{2}+25x-69 } $ with a remainder of $ \color{red}{ 238 } $.