Divide using synthetic division $$ ( 3x^{3}-25x^{2}+12x-32 ) \div ( x-8 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}8&3&-25&12&-32\\& & 24& -8& \color{black}{32} \\ \hline &\color{blue}{3}&\color{blue}{-1}&\color{blue}{4}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 3x^{3}-25x^{2}+12x-32 }{ x-8 } = \color{blue}{3x^{2}-x+4} $$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.
$$ \begin{array}{c|rrrr}\color{blue}{8}&3&-25&12&-32\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}8&\color{orangered}{ 3 }&-25&12&-32\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 3 } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrr}\color{blue}{8}&3&-25&12&-32\\& & \color{blue}{24} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -25 } + \color{orangered}{ 24 } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrr}8&3&\color{orangered}{ -25 }&12&-32\\& & \color{orangered}{24} & & \\ \hline &3&\color{orangered}{-1}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ -8 } $.
$$ \begin{array}{c|rrrr}\color{blue}{8}&3&-25&12&-32\\& & 24& \color{blue}{-8} & \\ \hline &3&\color{blue}{-1}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 12 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrr}8&3&-25&\color{orangered}{ 12 }&-32\\& & 24& \color{orangered}{-8} & \\ \hline &3&-1&\color{orangered}{4}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 4 } = \color{blue}{ 32 } $.
$$ \begin{array}{c|rrrr}\color{blue}{8}&3&-25&12&-32\\& & 24& -8& \color{blue}{32} \\ \hline &3&-1&\color{blue}{4}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -32 } + \color{orangered}{ 32 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}8&3&-25&12&\color{orangered}{ -32 }\\& & 24& -8& \color{orangered}{32} \\ \hline &\color{blue}{3}&\color{blue}{-1}&\color{blue}{4}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{2}-x+4 } $ with a remainder of $ \color{red}{ 0 } $.