Divide using synthetic division $$ ( 3x^{3}+5x^{2}-x+9 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}2&3&5&-1&9\\& & 6& 22& \color{black}{42} \\ \hline &\color{blue}{3}&\color{blue}{11}&\color{blue}{21}&\color{orangered}{51} \end{array} $$The solution is:
$$ \frac{ 3x^{3}+5x^{2}-x+9 }{ x-2 } = \color{blue}{3x^{2}+11x+21} ~+~ \frac{ \color{red}{ 51 } }{ x-2 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{2}&3&5&-1&9\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}2&\color{orangered}{ 3 }&5&-1&9\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 3 } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&3&5&-1&9\\& & \color{blue}{6} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 6 } = \color{orangered}{ 11 } $
$$ \begin{array}{c|rrrr}2&3&\color{orangered}{ 5 }&-1&9\\& & \color{orangered}{6} & & \\ \hline &3&\color{orangered}{11}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 11 } = \color{blue}{ 22 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&3&5&-1&9\\& & 6& \color{blue}{22} & \\ \hline &3&\color{blue}{11}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 22 } = \color{orangered}{ 21 } $
$$ \begin{array}{c|rrrr}2&3&5&\color{orangered}{ -1 }&9\\& & 6& \color{orangered}{22} & \\ \hline &3&11&\color{orangered}{21}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 21 } = \color{blue}{ 42 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&3&5&-1&9\\& & 6& 22& \color{blue}{42} \\ \hline &3&11&\color{blue}{21}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ 42 } = \color{orangered}{ 51 } $
$$ \begin{array}{c|rrrr}2&3&5&-1&\color{orangered}{ 9 }\\& & 6& 22& \color{orangered}{42} \\ \hline &\color{blue}{3}&\color{blue}{11}&\color{blue}{21}&\color{orangered}{51} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{2}+11x+21 } $ with a remainder of $ \color{red}{ 51 } $.