Divide using synthetic division $$ ( 2x^{6}+7x^{4}-x+1 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}3&2&0&7&0&0&-1&1\\& & 6& 18& 75& 225& 675& \color{black}{2022} \\ \hline &\color{blue}{2}&\color{blue}{6}&\color{blue}{25}&\color{blue}{75}&\color{blue}{225}&\color{blue}{674}&\color{orangered}{2023} \end{array} $$The solution is:
$$ \frac{ 2x^{6}+7x^{4}-x+1 }{ x-3 } = \color{blue}{2x^{5}+6x^{4}+25x^{3}+75x^{2}+225x+674} ~+~ \frac{ \color{red}{ 2023 } }{ x-3 } $$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.
$$ \begin{array}{c|rrrrrrr}\color{blue}{3}&2&0&7&0&0&-1&1\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}3&\color{orangered}{ 2 }&0&7&0&0&-1&1\\& & & & & & & \\ \hline &\color{orangered}{2}&&&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{3}&2&0&7&0&0&-1&1\\& & \color{blue}{6} & & & & & \\ \hline &\color{blue}{2}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 6 } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrrrrr}3&2&\color{orangered}{ 0 }&7&0&0&-1&1\\& & \color{orangered}{6} & & & & & \\ \hline &2&\color{orangered}{6}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 6 } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{3}&2&0&7&0&0&-1&1\\& & 6& \color{blue}{18} & & & & \\ \hline &2&\color{blue}{6}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 7 } + \color{orangered}{ 18 } = \color{orangered}{ 25 } $
$$ \begin{array}{c|rrrrrrr}3&2&0&\color{orangered}{ 7 }&0&0&-1&1\\& & 6& \color{orangered}{18} & & & & \\ \hline &2&6&\color{orangered}{25}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 25 } = \color{blue}{ 75 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{3}&2&0&7&0&0&-1&1\\& & 6& 18& \color{blue}{75} & & & \\ \hline &2&6&\color{blue}{25}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 75 } = \color{orangered}{ 75 } $
$$ \begin{array}{c|rrrrrrr}3&2&0&7&\color{orangered}{ 0 }&0&-1&1\\& & 6& 18& \color{orangered}{75} & & & \\ \hline &2&6&25&\color{orangered}{75}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 75 } = \color{blue}{ 225 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{3}&2&0&7&0&0&-1&1\\& & 6& 18& 75& \color{blue}{225} & & \\ \hline &2&6&25&\color{blue}{75}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 225 } = \color{orangered}{ 225 } $
$$ \begin{array}{c|rrrrrrr}3&2&0&7&0&\color{orangered}{ 0 }&-1&1\\& & 6& 18& 75& \color{orangered}{225} & & \\ \hline &2&6&25&75&\color{orangered}{225}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 225 } = \color{blue}{ 675 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{3}&2&0&7&0&0&-1&1\\& & 6& 18& 75& 225& \color{blue}{675} & \\ \hline &2&6&25&75&\color{blue}{225}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 675 } = \color{orangered}{ 674 } $
$$ \begin{array}{c|rrrrrrr}3&2&0&7&0&0&\color{orangered}{ -1 }&1\\& & 6& 18& 75& 225& \color{orangered}{675} & \\ \hline &2&6&25&75&225&\color{orangered}{674}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 674 } = \color{blue}{ 2022 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{3}&2&0&7&0&0&-1&1\\& & 6& 18& 75& 225& 675& \color{blue}{2022} \\ \hline &2&6&25&75&225&\color{blue}{674}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 2022 } = \color{orangered}{ 2023 } $
$$ \begin{array}{c|rrrrrrr}3&2&0&7&0&0&-1&\color{orangered}{ 1 }\\& & 6& 18& 75& 225& 675& \color{orangered}{2022} \\ \hline &\color{blue}{2}&\color{blue}{6}&\color{blue}{25}&\color{blue}{75}&\color{blue}{225}&\color{blue}{674}&\color{orangered}{2023} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{5}+6x^{4}+25x^{3}+75x^{2}+225x+674 } $ with a remainder of $ \color{red}{ 2023 } $.