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