Find remainder when $ x^{4}-8x^{3}+8x-58 $ is divided by $ x-8 $.

The synthetic division table is:

$$ \begin{array}{c|rrrrr}8&1&-8&0&8&-58\\& & 8& 0& 0& \color{black}{64} \\ \hline &\color{blue}{1}&\color{blue}{0}&\color{blue}{0}&\color{blue}{8}&\color{orangered}{6} \end{array} $$

The remainder when $ x^{4}-8x^{3}+8x-58 $ is divided by $ x-8 $ is $ \, \color{red}{ 6 } $.

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 -8 = 0 $ ( $ x = \color{blue}{ 8 } $ ) at the left.

$$ \begin{array}{c|rrrrr}\color{blue}{8}&1&-8&0&8&-58\\& & & & & \\ \hline &&&&& \end{array} $$

Step 1 : Bring down the leading coefficient to the bottom row.

$$ \begin{array}{c|rrrrr}8&\color{orangered}{ 1 }&-8&0&8&-58\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$

Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 1 } = \color{blue}{ 8 } $.

$$ \begin{array}{c|rrrrr}\color{blue}{8}&1&-8&0&8&-58\\& & \color{blue}{8} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$

Step 3 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ 8 } = \color{orangered}{ 0 } $

$$ \begin{array}{c|rrrrr}8&1&\color{orangered}{ -8 }&0&8&-58\\& & \color{orangered}{8} & & & \\ \hline &1&\color{orangered}{0}&&& \end{array} $$

Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.

$$ \begin{array}{c|rrrrr}\color{blue}{8}&1&-8&0&8&-58\\& & 8& \color{blue}{0} & & \\ \hline &1&\color{blue}{0}&&& \end{array} $$

Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $

$$ \begin{array}{c|rrrrr}8&1&-8&\color{orangered}{ 0 }&8&-58\\& & 8& \color{orangered}{0} & & \\ \hline &1&0&\color{orangered}{0}&& \end{array} $$

Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.

$$ \begin{array}{c|rrrrr}\color{blue}{8}&1&-8&0&8&-58\\& & 8& 0& \color{blue}{0} & \\ \hline &1&0&\color{blue}{0}&& \end{array} $$

Step 7 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ 0 } = \color{orangered}{ 8 } $

$$ \begin{array}{c|rrrrr}8&1&-8&0&\color{orangered}{ 8 }&-58\\& & 8& 0& \color{orangered}{0} & \\ \hline &1&0&0&\color{orangered}{8}& \end{array} $$

Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 8 } = \color{blue}{ 64 } $.

$$ \begin{array}{c|rrrrr}\color{blue}{8}&1&-8&0&8&-58\\& & 8& 0& 0& \color{blue}{64} \\ \hline &1&0&0&\color{blue}{8}& \end{array} $$

Step 9 : Add down last column: $ \color{orangered}{ -58 } + \color{orangered}{ 64 } = \color{orangered}{ 6 } $

$$ \begin{array}{c|rrrrr}8&1&-8&0&8&\color{orangered}{ -58 }\\& & 8& 0& 0& \color{orangered}{64} \\ \hline &\color{blue}{1}&\color{blue}{0}&\color{blue}{0}&\color{blue}{8}&\color{orangered}{6} \end{array} $$

Remainder is the last entry in the bottom row $ \left(\color{red}{ 6 }\right) $.

Synthetic Division Calculator