Divide using synthetic division $$ ( 3x^{8}-8x^{2}-12 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrrrr}3&3&0&0&0&0&0&-8&0&-12\\& & 9& 27& 81& 243& 729& 2187& 6537& \color{black}{19611} \\ \hline &\color{blue}{3}&\color{blue}{9}&\color{blue}{27}&\color{blue}{81}&\color{blue}{243}&\color{blue}{729}&\color{blue}{2179}&\color{blue}{6537}&\color{orangered}{19599} \end{array} $$The solution is:
$$ \frac{ 3x^{8}-8x^{2}-12 }{ x-3 } = \color{blue}{3x^{7}+9x^{6}+27x^{5}+81x^{4}+243x^{3}+729x^{2}+2179x+6537} ~+~ \frac{ \color{red}{ 19599 } }{ x-3 } $$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.
$$ \begin{array}{c|rrrrrrrrr}\color{blue}{3}&3&0&0&0&0&0&-8&0&-12\\& & & & & & & & & \\ \hline &&&&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrrrr}3&\color{orangered}{ 3 }&0&0&0&0&0&-8&0&-12\\& & & & & & & & & \\ \hline &\color{orangered}{3}&&&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 3 } = \color{blue}{ 9 } $.
$$ \begin{array}{c|rrrrrrrrr}\color{blue}{3}&3&0&0&0&0&0&-8&0&-12\\& & \color{blue}{9} & & & & & & & \\ \hline &\color{blue}{3}&&&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 9 } = \color{orangered}{ 9 } $
$$ \begin{array}{c|rrrrrrrrr}3&3&\color{orangered}{ 0 }&0&0&0&0&-8&0&-12\\& & \color{orangered}{9} & & & & & & & \\ \hline &3&\color{orangered}{9}&&&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 9 } = \color{blue}{ 27 } $.
$$ \begin{array}{c|rrrrrrrrr}\color{blue}{3}&3&0&0&0&0&0&-8&0&-12\\& & 9& \color{blue}{27} & & & & & & \\ \hline &3&\color{blue}{9}&&&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 27 } = \color{orangered}{ 27 } $
$$ \begin{array}{c|rrrrrrrrr}3&3&0&\color{orangered}{ 0 }&0&0&0&-8&0&-12\\& & 9& \color{orangered}{27} & & & & & & \\ \hline &3&9&\color{orangered}{27}&&&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 27 } = \color{blue}{ 81 } $.
$$ \begin{array}{c|rrrrrrrrr}\color{blue}{3}&3&0&0&0&0&0&-8&0&-12\\& & 9& 27& \color{blue}{81} & & & & & \\ \hline &3&9&\color{blue}{27}&&&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 81 } = \color{orangered}{ 81 } $
$$ \begin{array}{c|rrrrrrrrr}3&3&0&0&\color{orangered}{ 0 }&0&0&-8&0&-12\\& & 9& 27& \color{orangered}{81} & & & & & \\ \hline &3&9&27&\color{orangered}{81}&&&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 81 } = \color{blue}{ 243 } $.
$$ \begin{array}{c|rrrrrrrrr}\color{blue}{3}&3&0&0&0&0&0&-8&0&-12\\& & 9& 27& 81& \color{blue}{243} & & & & \\ \hline &3&9&27&\color{blue}{81}&&&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 243 } = \color{orangered}{ 243 } $
$$ \begin{array}{c|rrrrrrrrr}3&3&0&0&0&\color{orangered}{ 0 }&0&-8&0&-12\\& & 9& 27& 81& \color{orangered}{243} & & & & \\ \hline &3&9&27&81&\color{orangered}{243}&&&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 243 } = \color{blue}{ 729 } $.
$$ \begin{array}{c|rrrrrrrrr}\color{blue}{3}&3&0&0&0&0&0&-8&0&-12\\& & 9& 27& 81& 243& \color{blue}{729} & & & \\ \hline &3&9&27&81&\color{blue}{243}&&&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 729 } = \color{orangered}{ 729 } $
$$ \begin{array}{c|rrrrrrrrr}3&3&0&0&0&0&\color{orangered}{ 0 }&-8&0&-12\\& & 9& 27& 81& 243& \color{orangered}{729} & & & \\ \hline &3&9&27&81&243&\color{orangered}{729}&&& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 729 } = \color{blue}{ 2187 } $.
$$ \begin{array}{c|rrrrrrrrr}\color{blue}{3}&3&0&0&0&0&0&-8&0&-12\\& & 9& 27& 81& 243& 729& \color{blue}{2187} & & \\ \hline &3&9&27&81&243&\color{blue}{729}&&& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ 2187 } = \color{orangered}{ 2179 } $
$$ \begin{array}{c|rrrrrrrrr}3&3&0&0&0&0&0&\color{orangered}{ -8 }&0&-12\\& & 9& 27& 81& 243& 729& \color{orangered}{2187} & & \\ \hline &3&9&27&81&243&729&\color{orangered}{2179}&& \end{array} $$Step 14 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 2179 } = \color{blue}{ 6537 } $.
$$ \begin{array}{c|rrrrrrrrr}\color{blue}{3}&3&0&0&0&0&0&-8&0&-12\\& & 9& 27& 81& 243& 729& 2187& \color{blue}{6537} & \\ \hline &3&9&27&81&243&729&\color{blue}{2179}&& \end{array} $$Step 15 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 6537 } = \color{orangered}{ 6537 } $
$$ \begin{array}{c|rrrrrrrrr}3&3&0&0&0&0&0&-8&\color{orangered}{ 0 }&-12\\& & 9& 27& 81& 243& 729& 2187& \color{orangered}{6537} & \\ \hline &3&9&27&81&243&729&2179&\color{orangered}{6537}& \end{array} $$Step 16 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 6537 } = \color{blue}{ 19611 } $.
$$ \begin{array}{c|rrrrrrrrr}\color{blue}{3}&3&0&0&0&0&0&-8&0&-12\\& & 9& 27& 81& 243& 729& 2187& 6537& \color{blue}{19611} \\ \hline &3&9&27&81&243&729&2179&\color{blue}{6537}& \end{array} $$Step 17 : Add down last column: $ \color{orangered}{ -12 } + \color{orangered}{ 19611 } = \color{orangered}{ 19599 } $
$$ \begin{array}{c|rrrrrrrrr}3&3&0&0&0&0&0&-8&0&\color{orangered}{ -12 }\\& & 9& 27& 81& 243& 729& 2187& 6537& \color{orangered}{19611} \\ \hline &\color{blue}{3}&\color{blue}{9}&\color{blue}{27}&\color{blue}{81}&\color{blue}{243}&\color{blue}{729}&\color{blue}{2179}&\color{blue}{6537}&\color{orangered}{19599} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{7}+9x^{6}+27x^{5}+81x^{4}+243x^{3}+729x^{2}+2179x+6537 } $ with a remainder of $ \color{red}{ 19599 } $.