Divide using synthetic division $$ ( x^{4}+3x^{3}-13x^{2}+9 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}2&1&3&-13&0&9\\& & 2& 10& -6& \color{black}{-12} \\ \hline &\color{blue}{1}&\color{blue}{5}&\color{blue}{-3}&\color{blue}{-6}&\color{orangered}{-3} \end{array} $$The solution is:
$$ \frac{ x^{4}+3x^{3}-13x^{2}+9 }{ x-2 } = \color{blue}{x^{3}+5x^{2}-3x-6} \color{red}{~-~} \frac{ \color{red}{ 3 } }{ x-2 } $$Explanation
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.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&1&3&-13&0&9\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}2&\color{orangered}{ 1 }&3&-13&0&9\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 1 } = \color{blue}{ 2 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&1&3&-13&0&9\\& & \color{blue}{2} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 2 } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrrrr}2&1&\color{orangered}{ 3 }&-13&0&9\\& & \color{orangered}{2} & & & \\ \hline &1&\color{orangered}{5}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 5 } = \color{blue}{ 10 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&1&3&-13&0&9\\& & 2& \color{blue}{10} & & \\ \hline &1&\color{blue}{5}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -13 } + \color{orangered}{ 10 } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrrrr}2&1&3&\color{orangered}{ -13 }&0&9\\& & 2& \color{orangered}{10} & & \\ \hline &1&5&\color{orangered}{-3}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&1&3&-13&0&9\\& & 2& 10& \color{blue}{-6} & \\ \hline &1&5&\color{blue}{-3}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrrr}2&1&3&-13&\color{orangered}{ 0 }&9\\& & 2& 10& \color{orangered}{-6} & \\ \hline &1&5&-3&\color{orangered}{-6}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ -12 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&1&3&-13&0&9\\& & 2& 10& -6& \color{blue}{-12} \\ \hline &1&5&-3&\color{blue}{-6}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrrrr}2&1&3&-13&0&\color{orangered}{ 9 }\\& & 2& 10& -6& \color{orangered}{-12} \\ \hline &\color{blue}{1}&\color{blue}{5}&\color{blue}{-3}&\color{blue}{-6}&\color{orangered}{-3} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}+5x^{2}-3x-6 } $ with a remainder of $ \color{red}{ -3 } $.