Divide using synthetic division $$ ( -3x^{4}+2x^{3}+6x^{2}-5x+1 ) \div ( x-8 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}8&-3&2&6&-5&1\\& & -24& -176& -1360& \color{black}{-10920} \\ \hline &\color{blue}{-3}&\color{blue}{-22}&\color{blue}{-170}&\color{blue}{-1365}&\color{orangered}{-10919} \end{array} $$The solution is:
$$ \frac{ -3x^{4}+2x^{3}+6x^{2}-5x+1 }{ x-8 } = \color{blue}{-3x^{3}-22x^{2}-170x-1365} \color{red}{~-~} \frac{ \color{red}{ 10919 } }{ x-8 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{8}&-3&2&6&-5&1\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}8&\color{orangered}{ -3 }&2&6&-5&1\\& & & & & \\ \hline &\color{orangered}{-3}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ -24 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{8}&-3&2&6&-5&1\\& & \color{blue}{-24} & & & \\ \hline &\color{blue}{-3}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -24 \right) } = \color{orangered}{ -22 } $
$$ \begin{array}{c|rrrrr}8&-3&\color{orangered}{ 2 }&6&-5&1\\& & \color{orangered}{-24} & & & \\ \hline &-3&\color{orangered}{-22}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ \left( -22 \right) } = \color{blue}{ -176 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{8}&-3&2&6&-5&1\\& & -24& \color{blue}{-176} & & \\ \hline &-3&\color{blue}{-22}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ \left( -176 \right) } = \color{orangered}{ -170 } $
$$ \begin{array}{c|rrrrr}8&-3&2&\color{orangered}{ 6 }&-5&1\\& & -24& \color{orangered}{-176} & & \\ \hline &-3&-22&\color{orangered}{-170}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ \left( -170 \right) } = \color{blue}{ -1360 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{8}&-3&2&6&-5&1\\& & -24& -176& \color{blue}{-1360} & \\ \hline &-3&-22&\color{blue}{-170}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ \left( -1360 \right) } = \color{orangered}{ -1365 } $
$$ \begin{array}{c|rrrrr}8&-3&2&6&\color{orangered}{ -5 }&1\\& & -24& -176& \color{orangered}{-1360} & \\ \hline &-3&-22&-170&\color{orangered}{-1365}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ \left( -1365 \right) } = \color{blue}{ -10920 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{8}&-3&2&6&-5&1\\& & -24& -176& -1360& \color{blue}{-10920} \\ \hline &-3&-22&-170&\color{blue}{-1365}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ \left( -10920 \right) } = \color{orangered}{ -10919 } $
$$ \begin{array}{c|rrrrr}8&-3&2&6&-5&\color{orangered}{ 1 }\\& & -24& -176& -1360& \color{orangered}{-10920} \\ \hline &\color{blue}{-3}&\color{blue}{-22}&\color{blue}{-170}&\color{blue}{-1365}&\color{orangered}{-10919} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -3x^{3}-22x^{2}-170x-1365 } $ with a remainder of $ \color{red}{ -10919 } $.