Divide using synthetic division $$ ( x^{3}-9x^{2}-4x-65 ) \div ( x-10 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}10&1&-9&-4&-65\\& & 10& 10& \color{black}{60} \\ \hline &\color{blue}{1}&\color{blue}{1}&\color{blue}{6}&\color{orangered}{-5} \end{array} $$The solution is:
$$ \frac{ x^{3}-9x^{2}-4x-65 }{ x-10 } = \color{blue}{x^{2}+x+6} \color{red}{~-~} \frac{ \color{red}{ 5 } }{ x-10 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -10 = 0 $ ( $ x = \color{blue}{ 10 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{10}&1&-9&-4&-65\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}10&\color{orangered}{ 1 }&-9&-4&-65\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 10 } \cdot \color{blue}{ 1 } = \color{blue}{ 10 } $.
$$ \begin{array}{c|rrrr}\color{blue}{10}&1&-9&-4&-65\\& & \color{blue}{10} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ 10 } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrrr}10&1&\color{orangered}{ -9 }&-4&-65\\& & \color{orangered}{10} & & \\ \hline &1&\color{orangered}{1}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 10 } \cdot \color{blue}{ 1 } = \color{blue}{ 10 } $.
$$ \begin{array}{c|rrrr}\color{blue}{10}&1&-9&-4&-65\\& & 10& \color{blue}{10} & \\ \hline &1&\color{blue}{1}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 10 } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrr}10&1&-9&\color{orangered}{ -4 }&-65\\& & 10& \color{orangered}{10} & \\ \hline &1&1&\color{orangered}{6}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 10 } \cdot \color{blue}{ 6 } = \color{blue}{ 60 } $.
$$ \begin{array}{c|rrrr}\color{blue}{10}&1&-9&-4&-65\\& & 10& 10& \color{blue}{60} \\ \hline &1&1&\color{blue}{6}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -65 } + \color{orangered}{ 60 } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrrr}10&1&-9&-4&\color{orangered}{ -65 }\\& & 10& 10& \color{orangered}{60} \\ \hline &\color{blue}{1}&\color{blue}{1}&\color{blue}{6}&\color{orangered}{-5} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}+x+6 } $ with a remainder of $ \color{red}{ -5 } $.