Divide using synthetic division $$ ( x^{3}-5x^{2}-x-10 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}2&1&-5&-1&-10\\& & 2& -6& \color{black}{-14} \\ \hline &\color{blue}{1}&\color{blue}{-3}&\color{blue}{-7}&\color{orangered}{-24} \end{array} $$The solution is:
$$ \frac{ x^{3}-5x^{2}-x-10 }{ x-2 } = \color{blue}{x^{2}-3x-7} \color{red}{~-~} \frac{ \color{red}{ 24 } }{ x-2 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{2}&1&-5&-1&-10\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}2&\color{orangered}{ 1 }&-5&-1&-10\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 1 } = \color{blue}{ 2 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&1&-5&-1&-10\\& & \color{blue}{2} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ 2 } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrrr}2&1&\color{orangered}{ -5 }&-1&-10\\& & \color{orangered}{2} & & \\ \hline &1&\color{orangered}{-3}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&1&-5&-1&-10\\& & 2& \color{blue}{-6} & \\ \hline &1&\color{blue}{-3}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -7 } $
$$ \begin{array}{c|rrrr}2&1&-5&\color{orangered}{ -1 }&-10\\& & 2& \color{orangered}{-6} & \\ \hline &1&-3&\color{orangered}{-7}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -7 \right) } = \color{blue}{ -14 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&1&-5&-1&-10\\& & 2& -6& \color{blue}{-14} \\ \hline &1&-3&\color{blue}{-7}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ \left( -14 \right) } = \color{orangered}{ -24 } $
$$ \begin{array}{c|rrrr}2&1&-5&-1&\color{orangered}{ -10 }\\& & 2& -6& \color{orangered}{-14} \\ \hline &\color{blue}{1}&\color{blue}{-3}&\color{blue}{-7}&\color{orangered}{-24} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}-3x-7 } $ with a remainder of $ \color{red}{ -24 } $.