Divide using synthetic division $$ ( x^{3}-8x^{2}+x+42 ) \div ( x-7 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}7&1&-8&1&42\\& & 7& -7& \color{black}{-42} \\ \hline &\color{blue}{1}&\color{blue}{-1}&\color{blue}{-6}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{3}-8x^{2}+x+42 }{ x-7 } = \color{blue}{x^{2}-x-6} $$Explanation
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.
$$ \begin{array}{c|rrrr}\color{blue}{7}&1&-8&1&42\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}7&\color{orangered}{ 1 }&-8&1&42\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{7}&1&-8&1&42\\& & \color{blue}{7} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ 7 } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrr}7&1&\color{orangered}{ -8 }&1&42\\& & \color{orangered}{7} & & \\ \hline &1&\color{orangered}{-1}&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{7}&1&-8&1&42\\& & 7& \color{blue}{-7} & \\ \hline &1&\color{blue}{-1}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ \left( -7 \right) } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrr}7&1&-8&\color{orangered}{ 1 }&42\\& & 7& \color{orangered}{-7} & \\ \hline &1&-1&\color{orangered}{-6}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ -42 } $.
$$ \begin{array}{c|rrrr}\color{blue}{7}&1&-8&1&42\\& & 7& -7& \color{blue}{-42} \\ \hline &1&-1&\color{blue}{-6}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 42 } + \color{orangered}{ \left( -42 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}7&1&-8&1&\color{orangered}{ 42 }\\& & 7& -7& \color{orangered}{-42} \\ \hline &\color{blue}{1}&\color{blue}{-1}&\color{blue}{-6}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}-x-6 } $ with a remainder of $ \color{red}{ 0 } $.