Divide using synthetic division $$ ( x^{4}-x^{3}-12x^{2}-2x+8 ) \div ( x-4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}4&1&-1&-12&-2&8\\& & 4& 12& 0& \color{black}{-8} \\ \hline &\color{blue}{1}&\color{blue}{3}&\color{blue}{0}&\color{blue}{-2}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{4}-x^{3}-12x^{2}-2x+8 }{ x-4 } = \color{blue}{x^{3}+3x^{2}-2} $$Explanation
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.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&1&-1&-12&-2&8\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}4&\color{orangered}{ 1 }&-1&-12&-2&8\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 1 } = \color{blue}{ 4 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&1&-1&-12&-2&8\\& & \color{blue}{4} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 4 } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrrr}4&1&\color{orangered}{ -1 }&-12&-2&8\\& & \color{orangered}{4} & & & \\ \hline &1&\color{orangered}{3}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 3 } = \color{blue}{ 12 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&1&-1&-12&-2&8\\& & 4& \color{blue}{12} & & \\ \hline &1&\color{blue}{3}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -12 } + \color{orangered}{ 12 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}4&1&-1&\color{orangered}{ -12 }&-2&8\\& & 4& \color{orangered}{12} & & \\ \hline &1&3&\color{orangered}{0}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&1&-1&-12&-2&8\\& & 4& 12& \color{blue}{0} & \\ \hline &1&3&\color{blue}{0}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 0 } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrr}4&1&-1&-12&\color{orangered}{ -2 }&8\\& & 4& 12& \color{orangered}{0} & \\ \hline &1&3&0&\color{orangered}{-2}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ -8 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&1&-1&-12&-2&8\\& & 4& 12& 0& \color{blue}{-8} \\ \hline &1&3&0&\color{blue}{-2}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}4&1&-1&-12&-2&\color{orangered}{ 8 }\\& & 4& 12& 0& \color{orangered}{-8} \\ \hline &\color{blue}{1}&\color{blue}{3}&\color{blue}{0}&\color{blue}{-2}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}+3x^{2}-2 } $ with a remainder of $ \color{red}{ 0 } $.