Divide using synthetic division $$ ( 2x^{6}+9x^{5}-4x^{3}-x-9 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}3&2&9&0&-4&0&-1&-9\\& & 6& 45& 135& 393& 1179& \color{black}{3534} \\ \hline &\color{blue}{2}&\color{blue}{15}&\color{blue}{45}&\color{blue}{131}&\color{blue}{393}&\color{blue}{1178}&\color{orangered}{3525} \end{array} $$The solution is:
$$ \frac{ 2x^{6}+9x^{5}-4x^{3}-x-9 }{ x-3 } = \color{blue}{2x^{5}+15x^{4}+45x^{3}+131x^{2}+393x+1178} ~+~ \frac{ \color{red}{ 3525 } }{ x-3 } $$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.
$$ \begin{array}{c|rrrrrrr}\color{blue}{3}&2&9&0&-4&0&-1&-9\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}3&\color{orangered}{ 2 }&9&0&-4&0&-1&-9\\& & & & & & & \\ \hline &\color{orangered}{2}&&&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{3}&2&9&0&-4&0&-1&-9\\& & \color{blue}{6} & & & & & \\ \hline &\color{blue}{2}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ 6 } = \color{orangered}{ 15 } $
$$ \begin{array}{c|rrrrrrr}3&2&\color{orangered}{ 9 }&0&-4&0&-1&-9\\& & \color{orangered}{6} & & & & & \\ \hline &2&\color{orangered}{15}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 15 } = \color{blue}{ 45 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{3}&2&9&0&-4&0&-1&-9\\& & 6& \color{blue}{45} & & & & \\ \hline &2&\color{blue}{15}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 45 } = \color{orangered}{ 45 } $
$$ \begin{array}{c|rrrrrrr}3&2&9&\color{orangered}{ 0 }&-4&0&-1&-9\\& & 6& \color{orangered}{45} & & & & \\ \hline &2&15&\color{orangered}{45}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 45 } = \color{blue}{ 135 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{3}&2&9&0&-4&0&-1&-9\\& & 6& 45& \color{blue}{135} & & & \\ \hline &2&15&\color{blue}{45}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 135 } = \color{orangered}{ 131 } $
$$ \begin{array}{c|rrrrrrr}3&2&9&0&\color{orangered}{ -4 }&0&-1&-9\\& & 6& 45& \color{orangered}{135} & & & \\ \hline &2&15&45&\color{orangered}{131}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 131 } = \color{blue}{ 393 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{3}&2&9&0&-4&0&-1&-9\\& & 6& 45& 135& \color{blue}{393} & & \\ \hline &2&15&45&\color{blue}{131}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 393 } = \color{orangered}{ 393 } $
$$ \begin{array}{c|rrrrrrr}3&2&9&0&-4&\color{orangered}{ 0 }&-1&-9\\& & 6& 45& 135& \color{orangered}{393} & & \\ \hline &2&15&45&131&\color{orangered}{393}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 393 } = \color{blue}{ 1179 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{3}&2&9&0&-4&0&-1&-9\\& & 6& 45& 135& 393& \color{blue}{1179} & \\ \hline &2&15&45&131&\color{blue}{393}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 1179 } = \color{orangered}{ 1178 } $
$$ \begin{array}{c|rrrrrrr}3&2&9&0&-4&0&\color{orangered}{ -1 }&-9\\& & 6& 45& 135& 393& \color{orangered}{1179} & \\ \hline &2&15&45&131&393&\color{orangered}{1178}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 1178 } = \color{blue}{ 3534 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{3}&2&9&0&-4&0&-1&-9\\& & 6& 45& 135& 393& 1179& \color{blue}{3534} \\ \hline &2&15&45&131&393&\color{blue}{1178}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ 3534 } = \color{orangered}{ 3525 } $
$$ \begin{array}{c|rrrrrrr}3&2&9&0&-4&0&-1&\color{orangered}{ -9 }\\& & 6& 45& 135& 393& 1179& \color{orangered}{3534} \\ \hline &\color{blue}{2}&\color{blue}{15}&\color{blue}{45}&\color{blue}{131}&\color{blue}{393}&\color{blue}{1178}&\color{orangered}{3525} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{5}+15x^{4}+45x^{3}+131x^{2}+393x+1178 } $ with a remainder of $ \color{red}{ 3525 } $.