Divide using synthetic division $$ ( -6x^{3}-7x^{2}+9 ) \div ( -7 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}0&-6&-7&0&9\\& & 0& 0& \color{black}{0} \\ \hline &\color{blue}{-6}&\color{blue}{-7}&\color{blue}{0}&\color{orangered}{9} \end{array} $$The solution is:
$$ \frac{ -6x^{3}-7x^{2}+9 }{ x } = \color{blue}{-6x^{2}-7x} ~+~ \frac{ \color{red}{ 9 } }{ x } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.
$$ \begin{array}{c|rrrr}\color{blue}{0}&-6&-7&0&9\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}0&\color{orangered}{ -6 }&-7&0&9\\& & & & \\ \hline &\color{orangered}{-6}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrr}\color{blue}{0}&-6&-7&0&9\\& & \color{blue}{0} & & \\ \hline &\color{blue}{-6}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ 0 } = \color{orangered}{ -7 } $
$$ \begin{array}{c|rrrr}0&-6&\color{orangered}{ -7 }&0&9\\& & \color{orangered}{0} & & \\ \hline &-6&\color{orangered}{-7}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -7 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrr}\color{blue}{0}&-6&-7&0&9\\& & 0& \color{blue}{0} & \\ \hline &-6&\color{blue}{-7}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}0&-6&-7&\color{orangered}{ 0 }&9\\& & 0& \color{orangered}{0} & \\ \hline &-6&-7&\color{orangered}{0}& \end{array} $$Step 6 : 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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{0}&-6&-7&0&9\\& & 0& 0& \color{blue}{0} \\ \hline &-6&-7&\color{blue}{0}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ 0 } = \color{orangered}{ 9 } $
$$ \begin{array}{c|rrrr}0&-6&-7&0&\color{orangered}{ 9 }\\& & 0& 0& \color{orangered}{0} \\ \hline &\color{blue}{-6}&\color{blue}{-7}&\color{blue}{0}&\color{orangered}{9} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -6x^{2}-7x } $ with a remainder of $ \color{red}{ 9 } $.