Find remainder when $ 6x^{6}-4x^{4}+3x^{2}+3 $ is divided by $ x-2 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}2&6&0&-4&0&3&0&3\\& & 12& 24& 40& 80& 166& \color{black}{332} \\ \hline &\color{blue}{6}&\color{blue}{12}&\color{blue}{20}&\color{blue}{40}&\color{blue}{83}&\color{blue}{166}&\color{orangered}{335} \end{array} $$The remainder when $ 6x^{6}-4x^{4}+3x^{2}+3 $ is divided by $ x-2 $ is $ \, \color{red}{ 335 } $.
Explanation
We can find remainder using synthetic division method.
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -2 = 0 $ ( $ x = \color{blue}{ 2 } $ ) at the left.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&6&0&-4&0&3&0&3\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}2&\color{orangered}{ 6 }&0&-4&0&3&0&3\\& & & & & & & \\ \hline &\color{orangered}{6}&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 6 } = \color{blue}{ 12 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&6&0&-4&0&3&0&3\\& & \color{blue}{12} & & & & & \\ \hline &\color{blue}{6}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 12 } = \color{orangered}{ 12 } $
$$ \begin{array}{c|rrrrrrr}2&6&\color{orangered}{ 0 }&-4&0&3&0&3\\& & \color{orangered}{12} & & & & & \\ \hline &6&\color{orangered}{12}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 12 } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&6&0&-4&0&3&0&3\\& & 12& \color{blue}{24} & & & & \\ \hline &6&\color{blue}{12}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 24 } = \color{orangered}{ 20 } $
$$ \begin{array}{c|rrrrrrr}2&6&0&\color{orangered}{ -4 }&0&3&0&3\\& & 12& \color{orangered}{24} & & & & \\ \hline &6&12&\color{orangered}{20}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 20 } = \color{blue}{ 40 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&6&0&-4&0&3&0&3\\& & 12& 24& \color{blue}{40} & & & \\ \hline &6&12&\color{blue}{20}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 40 } = \color{orangered}{ 40 } $
$$ \begin{array}{c|rrrrrrr}2&6&0&-4&\color{orangered}{ 0 }&3&0&3\\& & 12& 24& \color{orangered}{40} & & & \\ \hline &6&12&20&\color{orangered}{40}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 40 } = \color{blue}{ 80 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&6&0&-4&0&3&0&3\\& & 12& 24& 40& \color{blue}{80} & & \\ \hline &6&12&20&\color{blue}{40}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 80 } = \color{orangered}{ 83 } $
$$ \begin{array}{c|rrrrrrr}2&6&0&-4&0&\color{orangered}{ 3 }&0&3\\& & 12& 24& 40& \color{orangered}{80} & & \\ \hline &6&12&20&40&\color{orangered}{83}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 83 } = \color{blue}{ 166 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&6&0&-4&0&3&0&3\\& & 12& 24& 40& 80& \color{blue}{166} & \\ \hline &6&12&20&40&\color{blue}{83}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 166 } = \color{orangered}{ 166 } $
$$ \begin{array}{c|rrrrrrr}2&6&0&-4&0&3&\color{orangered}{ 0 }&3\\& & 12& 24& 40& 80& \color{orangered}{166} & \\ \hline &6&12&20&40&83&\color{orangered}{166}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 166 } = \color{blue}{ 332 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&6&0&-4&0&3&0&3\\& & 12& 24& 40& 80& 166& \color{blue}{332} \\ \hline &6&12&20&40&83&\color{blue}{166}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 332 } = \color{orangered}{ 335 } $
$$ \begin{array}{c|rrrrrrr}2&6&0&-4&0&3&0&\color{orangered}{ 3 }\\& & 12& 24& 40& 80& 166& \color{orangered}{332} \\ \hline &\color{blue}{6}&\color{blue}{12}&\color{blue}{20}&\color{blue}{40}&\color{blue}{83}&\color{blue}{166}&\color{orangered}{335} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 335 }\right) $.