Divide using synthetic division $$ ( -12x^{2}-5 ) \div ( 3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}0&-12&0&-5\\& & 0& \color{black}{0} \\ \hline &\color{blue}{-12}&\color{blue}{0}&\color{orangered}{-5} \end{array} $$The solution is:
$$ \frac{ -12x^{2}-5 }{ x } = \color{blue}{-12x} \color{red}{~-~} \frac{ \color{red}{ 5 } }{ x } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.
$$ \begin{array}{c|rrr}\color{blue}{0}&-12&0&-5\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}0&\color{orangered}{ -12 }&0&-5\\& & & \\ \hline &\color{orangered}{-12}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -12 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrr}\color{blue}{0}&-12&0&-5\\& & \color{blue}{0} & \\ \hline &\color{blue}{-12}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrr}0&-12&\color{orangered}{ 0 }&-5\\& & \color{orangered}{0} & \\ \hline &-12&\color{orangered}{0}& \end{array} $$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 } $.
$$ \begin{array}{c|rrr}\color{blue}{0}&-12&0&-5\\& & 0& \color{blue}{0} \\ \hline &-12&\color{blue}{0}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ 0 } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrr}0&-12&0&\color{orangered}{ -5 }\\& & 0& \color{orangered}{0} \\ \hline &\color{blue}{-12}&\color{blue}{0}&\color{orangered}{-5} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -12x } $ with a remainder of $ \color{red}{ -5 } $.