Divide using synthetic division $$ ( -x^{3}+10x^{2}-8x-64 ) \div ( x-8 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}8&-1&10&-8&-64\\& & -8& 16& \color{black}{64} \\ \hline &\color{blue}{-1}&\color{blue}{2}&\color{blue}{8}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ -x^{3}+10x^{2}-8x-64 }{ x-8 } = \color{blue}{-x^{2}+2x+8} $$Explanation
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.
$$ \begin{array}{c|rrrr}\color{blue}{8}&-1&10&-8&-64\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}8&\color{orangered}{ -1 }&10&-8&-64\\& & & & \\ \hline &\color{orangered}{-1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ -8 } $.
$$ \begin{array}{c|rrrr}\color{blue}{8}&-1&10&-8&-64\\& & \color{blue}{-8} & & \\ \hline &\color{blue}{-1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 10 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrr}8&-1&\color{orangered}{ 10 }&-8&-64\\& & \color{orangered}{-8} & & \\ \hline &-1&\color{orangered}{2}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 2 } = \color{blue}{ 16 } $.
$$ \begin{array}{c|rrrr}\color{blue}{8}&-1&10&-8&-64\\& & -8& \color{blue}{16} & \\ \hline &-1&\color{blue}{2}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ 16 } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrr}8&-1&10&\color{orangered}{ -8 }&-64\\& & -8& \color{orangered}{16} & \\ \hline &-1&2&\color{orangered}{8}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 8 } = \color{blue}{ 64 } $.
$$ \begin{array}{c|rrrr}\color{blue}{8}&-1&10&-8&-64\\& & -8& 16& \color{blue}{64} \\ \hline &-1&2&\color{blue}{8}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -64 } + \color{orangered}{ 64 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}8&-1&10&-8&\color{orangered}{ -64 }\\& & -8& 16& \color{orangered}{64} \\ \hline &\color{blue}{-1}&\color{blue}{2}&\color{blue}{8}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -x^{2}+2x+8 } $ with a remainder of $ \color{red}{ 0 } $.