Divide using synthetic division $$ ( -9x^{2}+35 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}3&-9&0&35\\& & -27& \color{black}{-81} \\ \hline &\color{blue}{-9}&\color{blue}{-27}&\color{orangered}{-46} \end{array} $$The solution is:
$$ \frac{ -9x^{2}+35 }{ x-3 } = \color{blue}{-9x-27} \color{red}{~-~} \frac{ \color{red}{ 46 } }{ x-3 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -3 = 0 $ ( $ x = \color{blue}{ 3 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{3}&-9&0&35\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}3&\color{orangered}{ -9 }&0&35\\& & & \\ \hline &\color{orangered}{-9}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ -27 } $.
$$ \begin{array}{c|rrr}\color{blue}{3}&-9&0&35\\& & \color{blue}{-27} & \\ \hline &\color{blue}{-9}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -27 \right) } = \color{orangered}{ -27 } $
$$ \begin{array}{c|rrr}3&-9&\color{orangered}{ 0 }&35\\& & \color{orangered}{-27} & \\ \hline &-9&\color{orangered}{-27}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -27 \right) } = \color{blue}{ -81 } $.
$$ \begin{array}{c|rrr}\color{blue}{3}&-9&0&35\\& & -27& \color{blue}{-81} \\ \hline &-9&\color{blue}{-27}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 35 } + \color{orangered}{ \left( -81 \right) } = \color{orangered}{ -46 } $
$$ \begin{array}{c|rrr}3&-9&0&\color{orangered}{ 35 }\\& & -27& \color{orangered}{-81} \\ \hline &\color{blue}{-9}&\color{blue}{-27}&\color{orangered}{-46} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -9x-27 } $ with a remainder of $ \color{red}{ -46 } $.