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