Divide using synthetic division $$ ( 2x^{6}-18x^{4}+x^{2}-9 ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}-3&2&0&-18&0&1&0&-9\\& & -6& 18& 0& 0& -3& \color{black}{9} \\ \hline &\color{blue}{2}&\color{blue}{-6}&\color{blue}{0}&\color{blue}{0}&\color{blue}{1}&\color{blue}{-3}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 2x^{6}-18x^{4}+x^{2}-9 }{ x+3 } = \color{blue}{2x^{5}-6x^{4}+x-3} $$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.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-3}&2&0&-18&0&1&0&-9\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}-3&\color{orangered}{ 2 }&0&-18&0&1&0&-9\\& & & & & & & \\ \hline &\color{orangered}{2}&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 2 } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-3}&2&0&-18&0&1&0&-9\\& & \color{blue}{-6} & & & & & \\ \hline &\color{blue}{2}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrrrrr}-3&2&\color{orangered}{ 0 }&-18&0&1&0&-9\\& & \color{orangered}{-6} & & & & & \\ \hline &2&\color{orangered}{-6}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-3}&2&0&-18&0&1&0&-9\\& & -6& \color{blue}{18} & & & & \\ \hline &2&\color{blue}{-6}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -18 } + \color{orangered}{ 18 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrrr}-3&2&0&\color{orangered}{ -18 }&0&1&0&-9\\& & -6& \color{orangered}{18} & & & & \\ \hline &2&-6&\color{orangered}{0}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-3}&2&0&-18&0&1&0&-9\\& & -6& 18& \color{blue}{0} & & & \\ \hline &2&-6&\color{blue}{0}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrrr}-3&2&0&-18&\color{orangered}{ 0 }&1&0&-9\\& & -6& 18& \color{orangered}{0} & & & \\ \hline &2&-6&0&\color{orangered}{0}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-3}&2&0&-18&0&1&0&-9\\& & -6& 18& 0& \color{blue}{0} & & \\ \hline &2&-6&0&\color{blue}{0}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 0 } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrrrrrr}-3&2&0&-18&0&\color{orangered}{ 1 }&0&-9\\& & -6& 18& 0& \color{orangered}{0} & & \\ \hline &2&-6&0&0&\color{orangered}{1}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 1 } = \color{blue}{ -3 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-3}&2&0&-18&0&1&0&-9\\& & -6& 18& 0& 0& \color{blue}{-3} & \\ \hline &2&-6&0&0&\color{blue}{1}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -3 \right) } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrrrrrr}-3&2&0&-18&0&1&\color{orangered}{ 0 }&-9\\& & -6& 18& 0& 0& \color{orangered}{-3} & \\ \hline &2&-6&0&0&1&\color{orangered}{-3}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ 9 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-3}&2&0&-18&0&1&0&-9\\& & -6& 18& 0& 0& -3& \color{blue}{9} \\ \hline &2&-6&0&0&1&\color{blue}{-3}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ 9 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrrr}-3&2&0&-18&0&1&0&\color{orangered}{ -9 }\\& & -6& 18& 0& 0& -3& \color{orangered}{9} \\ \hline &\color{blue}{2}&\color{blue}{-6}&\color{blue}{0}&\color{blue}{0}&\color{blue}{1}&\color{blue}{-3}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{5}-6x^{4}+x-3 } $ with a remainder of $ \color{red}{ 0 } $.