Find remainder when $ 3x^{4}+4x^{2}-30x+20 $ is divided by $ x-2 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}2&3&0&4&-30&20\\& & 6& 12& 32& \color{black}{4} \\ \hline &\color{blue}{3}&\color{blue}{6}&\color{blue}{16}&\color{blue}{2}&\color{orangered}{24} \end{array} $$The remainder when $ 3x^{4}+4x^{2}-30x+20 $ is divided by $ x-2 $ is $ \, \color{red}{ 24 } $.
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.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&3&0&4&-30&20\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}2&\color{orangered}{ 3 }&0&4&-30&20\\& & & & & \\ \hline &\color{orangered}{3}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 3 } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&3&0&4&-30&20\\& & \color{blue}{6} & & & \\ \hline &\color{blue}{3}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 6 } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrrr}2&3&\color{orangered}{ 0 }&4&-30&20\\& & \color{orangered}{6} & & & \\ \hline &3&\color{orangered}{6}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 6 } = \color{blue}{ 12 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&3&0&4&-30&20\\& & 6& \color{blue}{12} & & \\ \hline &3&\color{blue}{6}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 12 } = \color{orangered}{ 16 } $
$$ \begin{array}{c|rrrrr}2&3&0&\color{orangered}{ 4 }&-30&20\\& & 6& \color{orangered}{12} & & \\ \hline &3&6&\color{orangered}{16}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 16 } = \color{blue}{ 32 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&3&0&4&-30&20\\& & 6& 12& \color{blue}{32} & \\ \hline &3&6&\color{blue}{16}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -30 } + \color{orangered}{ 32 } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrrr}2&3&0&4&\color{orangered}{ -30 }&20\\& & 6& 12& \color{orangered}{32} & \\ \hline &3&6&16&\color{orangered}{2}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 2 } = \color{blue}{ 4 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&3&0&4&-30&20\\& & 6& 12& 32& \color{blue}{4} \\ \hline &3&6&16&\color{blue}{2}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 20 } + \color{orangered}{ 4 } = \color{orangered}{ 24 } $
$$ \begin{array}{c|rrrrr}2&3&0&4&-30&\color{orangered}{ 20 }\\& & 6& 12& 32& \color{orangered}{4} \\ \hline &\color{blue}{3}&\color{blue}{6}&\color{blue}{16}&\color{blue}{2}&\color{orangered}{24} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 24 }\right) $.