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