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