Divide using synthetic division $$ ( x^{3}-6x^{2}+14x-9 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}3&1&-6&14&-9\\& & 3& -9& \color{black}{15} \\ \hline &\color{blue}{1}&\color{blue}{-3}&\color{blue}{5}&\color{orangered}{6} \end{array} $$The solution is:
$$ \frac{ x^{3}-6x^{2}+14x-9 }{ x-3 } = \color{blue}{x^{2}-3x+5} ~+~ \frac{ \color{red}{ 6 } }{ x-3 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{3}&1&-6&14&-9\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}3&\color{orangered}{ 1 }&-6&14&-9\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 1 } = \color{blue}{ 3 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&1&-6&14&-9\\& & \color{blue}{3} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ 3 } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrrr}3&1&\color{orangered}{ -6 }&14&-9\\& & \color{orangered}{3} & & \\ \hline &1&\color{orangered}{-3}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ -9 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&1&-6&14&-9\\& & 3& \color{blue}{-9} & \\ \hline &1&\color{blue}{-3}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 14 } + \color{orangered}{ \left( -9 \right) } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrrr}3&1&-6&\color{orangered}{ 14 }&-9\\& & 3& \color{orangered}{-9} & \\ \hline &1&-3&\color{orangered}{5}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 5 } = \color{blue}{ 15 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&1&-6&14&-9\\& & 3& -9& \color{blue}{15} \\ \hline &1&-3&\color{blue}{5}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ 15 } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrr}3&1&-6&14&\color{orangered}{ -9 }\\& & 3& -9& \color{orangered}{15} \\ \hline &\color{blue}{1}&\color{blue}{-3}&\color{blue}{5}&\color{orangered}{6} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}-3x+5 } $ with a remainder of $ \color{red}{ 6 } $.