Find remainder when $ 2x^{4}-34x^{2}+4x-25 $ is divided by $ x-4 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}4&2&0&-34&4&-25\\& & 8& 32& -8& \color{black}{-16} \\ \hline &\color{blue}{2}&\color{blue}{8}&\color{blue}{-2}&\color{blue}{-4}&\color{orangered}{-41} \end{array} $$The remainder when $ 2x^{4}-34x^{2}+4x-25 $ is divided by $ x-4 $ is $ \, \color{red}{ -41 } $.
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 -4 = 0 $ ( $ x = \color{blue}{ 4 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&2&0&-34&4&-25\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}4&\color{orangered}{ 2 }&0&-34&4&-25\\& & & & & \\ \hline &\color{orangered}{2}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 2 } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&2&0&-34&4&-25\\& & \color{blue}{8} & & & \\ \hline &\color{blue}{2}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 8 } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrrr}4&2&\color{orangered}{ 0 }&-34&4&-25\\& & \color{orangered}{8} & & & \\ \hline &2&\color{orangered}{8}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 8 } = \color{blue}{ 32 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&2&0&-34&4&-25\\& & 8& \color{blue}{32} & & \\ \hline &2&\color{blue}{8}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -34 } + \color{orangered}{ 32 } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrr}4&2&0&\color{orangered}{ -34 }&4&-25\\& & 8& \color{orangered}{32} & & \\ \hline &2&8&\color{orangered}{-2}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ -8 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&2&0&-34&4&-25\\& & 8& 32& \color{blue}{-8} & \\ \hline &2&8&\color{blue}{-2}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrrr}4&2&0&-34&\color{orangered}{ 4 }&-25\\& & 8& 32& \color{orangered}{-8} & \\ \hline &2&8&-2&\color{orangered}{-4}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ -16 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&2&0&-34&4&-25\\& & 8& 32& -8& \color{blue}{-16} \\ \hline &2&8&-2&\color{blue}{-4}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -25 } + \color{orangered}{ \left( -16 \right) } = \color{orangered}{ -41 } $
$$ \begin{array}{c|rrrrr}4&2&0&-34&4&\color{orangered}{ -25 }\\& & 8& 32& -8& \color{orangered}{-16} \\ \hline &\color{blue}{2}&\color{blue}{8}&\color{blue}{-2}&\color{blue}{-4}&\color{orangered}{-41} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -41 }\right) $.