Find remainder when $ x^{4}-8x^{3}-x^{2}+62x-34 $ is divided by $ x-7 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}7&1&-8&-1&62&-34\\& & 7& -7& -56& \color{black}{42} \\ \hline &\color{blue}{1}&\color{blue}{-1}&\color{blue}{-8}&\color{blue}{6}&\color{orangered}{8} \end{array} $$The remainder when $ x^{4}-8x^{3}-x^{2}+62x-34 $ is divided by $ x-7 $ is $ \, \color{red}{ 8 } $.
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 -7 = 0 $ ( $ x = \color{blue}{ 7 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{7}&1&-8&-1&62&-34\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}7&\color{orangered}{ 1 }&-8&-1&62&-34\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ 1 } = \color{blue}{ 7 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{7}&1&-8&-1&62&-34\\& & \color{blue}{7} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ 7 } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrrr}7&1&\color{orangered}{ -8 }&-1&62&-34\\& & \color{orangered}{7} & & & \\ \hline &1&\color{orangered}{-1}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ -7 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{7}&1&-8&-1&62&-34\\& & 7& \color{blue}{-7} & & \\ \hline &1&\color{blue}{-1}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ \left( -7 \right) } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrrrr}7&1&-8&\color{orangered}{ -1 }&62&-34\\& & 7& \color{orangered}{-7} & & \\ \hline &1&-1&\color{orangered}{-8}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ -56 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{7}&1&-8&-1&62&-34\\& & 7& -7& \color{blue}{-56} & \\ \hline &1&-1&\color{blue}{-8}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 62 } + \color{orangered}{ \left( -56 \right) } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrrr}7&1&-8&-1&\color{orangered}{ 62 }&-34\\& & 7& -7& \color{orangered}{-56} & \\ \hline &1&-1&-8&\color{orangered}{6}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ 6 } = \color{blue}{ 42 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{7}&1&-8&-1&62&-34\\& & 7& -7& -56& \color{blue}{42} \\ \hline &1&-1&-8&\color{blue}{6}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -34 } + \color{orangered}{ 42 } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrrr}7&1&-8&-1&62&\color{orangered}{ -34 }\\& & 7& -7& -56& \color{orangered}{42} \\ \hline &\color{blue}{1}&\color{blue}{-1}&\color{blue}{-8}&\color{blue}{6}&\color{orangered}{8} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 8 }\right) $.