Divide using synthetic division $$ ( x^{3}-729 ) \div ( x-9 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}9&1&0&0&-729\\& & 9& 81& \color{black}{729} \\ \hline &\color{blue}{1}&\color{blue}{9}&\color{blue}{81}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{3}-729 }{ x-9 } = \color{blue}{x^{2}+9x+81} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -9 = 0 $ ( $ x = \color{blue}{ 9 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{9}&1&0&0&-729\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}9&\color{orangered}{ 1 }&0&0&-729\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ 1 } = \color{blue}{ 9 } $.
$$ \begin{array}{c|rrrr}\color{blue}{9}&1&0&0&-729\\& & \color{blue}{9} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 9 } = \color{orangered}{ 9 } $
$$ \begin{array}{c|rrrr}9&1&\color{orangered}{ 0 }&0&-729\\& & \color{orangered}{9} & & \\ \hline &1&\color{orangered}{9}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ 9 } = \color{blue}{ 81 } $.
$$ \begin{array}{c|rrrr}\color{blue}{9}&1&0&0&-729\\& & 9& \color{blue}{81} & \\ \hline &1&\color{blue}{9}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 81 } = \color{orangered}{ 81 } $
$$ \begin{array}{c|rrrr}9&1&0&\color{orangered}{ 0 }&-729\\& & 9& \color{orangered}{81} & \\ \hline &1&9&\color{orangered}{81}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ 81 } = \color{blue}{ 729 } $.
$$ \begin{array}{c|rrrr}\color{blue}{9}&1&0&0&-729\\& & 9& 81& \color{blue}{729} \\ \hline &1&9&\color{blue}{81}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -729 } + \color{orangered}{ 729 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}9&1&0&0&\color{orangered}{ -729 }\\& & 9& 81& \color{orangered}{729} \\ \hline &\color{blue}{1}&\color{blue}{9}&\color{blue}{81}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}+9x+81 } $ with a remainder of $ \color{red}{ 0 } $.