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