Divide using synthetic division $$ ( -64x^{3}+4x-32 ) \div ( -8 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}0&-64&0&4&-32\\& & 0& 0& \color{black}{0} \\ \hline &\color{blue}{-64}&\color{blue}{0}&\color{blue}{4}&\color{orangered}{-32} \end{array} $$The solution is:
$$ \frac{ -64x^{3}+4x-32 }{ x } = \color{blue}{-64x^{2}+4} \color{red}{~-~} \frac{ \color{red}{ 32 } }{ x } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.
$$ \begin{array}{c|rrrr}\color{blue}{0}&-64&0&4&-32\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}0&\color{orangered}{ -64 }&0&4&-32\\& & & & \\ \hline &\color{orangered}{-64}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -64 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrr}\color{blue}{0}&-64&0&4&-32\\& & \color{blue}{0} & & \\ \hline &\color{blue}{-64}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}0&-64&\color{orangered}{ 0 }&4&-32\\& & \color{orangered}{0} & & \\ \hline &-64&\color{orangered}{0}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrr}\color{blue}{0}&-64&0&4&-32\\& & 0& \color{blue}{0} & \\ \hline &-64&\color{blue}{0}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 0 } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrr}0&-64&0&\color{orangered}{ 4 }&-32\\& & 0& \color{orangered}{0} & \\ \hline &-64&0&\color{orangered}{4}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 4 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrr}\color{blue}{0}&-64&0&4&-32\\& & 0& 0& \color{blue}{0} \\ \hline &-64&0&\color{blue}{4}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -32 } + \color{orangered}{ 0 } = \color{orangered}{ -32 } $
$$ \begin{array}{c|rrrr}0&-64&0&4&\color{orangered}{ -32 }\\& & 0& 0& \color{orangered}{0} \\ \hline &\color{blue}{-64}&\color{blue}{0}&\color{blue}{4}&\color{orangered}{-32} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -64x^{2}+4 } $ with a remainder of $ \color{red}{ -32 } $.