Find remainder when $ x^{2}-74 $ is divided by $ x-8 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}8&1&0&-74\\& & 8& \color{black}{64} \\ \hline &\color{blue}{1}&\color{blue}{8}&\color{orangered}{-10} \end{array} $$The remainder when $ x^{2}-74 $ is divided by $ x-8 $ is $ \, \color{red}{ -10 } $.
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 -8 = 0 $ ( $ x = \color{blue}{ 8 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{8}&1&0&-74\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}8&\color{orangered}{ 1 }&0&-74\\& & & \\ \hline &\color{orangered}{1}&& \end{array} $$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 } $.
$$ \begin{array}{c|rrr}\color{blue}{8}&1&0&-74\\& & \color{blue}{8} & \\ \hline &\color{blue}{1}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 8 } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrr}8&1&\color{orangered}{ 0 }&-74\\& & \color{orangered}{8} & \\ \hline &1&\color{orangered}{8}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 8 } = \color{blue}{ 64 } $.
$$ \begin{array}{c|rrr}\color{blue}{8}&1&0&-74\\& & 8& \color{blue}{64} \\ \hline &1&\color{blue}{8}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -74 } + \color{orangered}{ 64 } = \color{orangered}{ -10 } $
$$ \begin{array}{c|rrr}8&1&0&\color{orangered}{ -74 }\\& & 8& \color{orangered}{64} \\ \hline &\color{blue}{1}&\color{blue}{8}&\color{orangered}{-10} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -10 }\right) $.