Divide using synthetic division $$ ( x^{4}-12x^{3}+43x^{2}-112x+192 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}3&1&-12&43&-112&192\\& & 3& -27& 48& \color{black}{-192} \\ \hline &\color{blue}{1}&\color{blue}{-9}&\color{blue}{16}&\color{blue}{-64}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{4}-12x^{3}+43x^{2}-112x+192 }{ x-3 } = \color{blue}{x^{3}-9x^{2}+16x-64} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -3 = 0 $ ( $ x = \color{blue}{ 3 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&1&-12&43&-112&192\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}3&\color{orangered}{ 1 }&-12&43&-112&192\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 1 } = \color{blue}{ 3 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&1&-12&43&-112&192\\& & \color{blue}{3} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -12 } + \color{orangered}{ 3 } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrrr}3&1&\color{orangered}{ -12 }&43&-112&192\\& & \color{orangered}{3} & & & \\ \hline &1&\color{orangered}{-9}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ -27 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&1&-12&43&-112&192\\& & 3& \color{blue}{-27} & & \\ \hline &1&\color{blue}{-9}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 43 } + \color{orangered}{ \left( -27 \right) } = \color{orangered}{ 16 } $
$$ \begin{array}{c|rrrrr}3&1&-12&\color{orangered}{ 43 }&-112&192\\& & 3& \color{orangered}{-27} & & \\ \hline &1&-9&\color{orangered}{16}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 16 } = \color{blue}{ 48 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&1&-12&43&-112&192\\& & 3& -27& \color{blue}{48} & \\ \hline &1&-9&\color{blue}{16}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -112 } + \color{orangered}{ 48 } = \color{orangered}{ -64 } $
$$ \begin{array}{c|rrrrr}3&1&-12&43&\color{orangered}{ -112 }&192\\& & 3& -27& \color{orangered}{48} & \\ \hline &1&-9&16&\color{orangered}{-64}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -64 \right) } = \color{blue}{ -192 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&1&-12&43&-112&192\\& & 3& -27& 48& \color{blue}{-192} \\ \hline &1&-9&16&\color{blue}{-64}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 192 } + \color{orangered}{ \left( -192 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}3&1&-12&43&-112&\color{orangered}{ 192 }\\& & 3& -27& 48& \color{orangered}{-192} \\ \hline &\color{blue}{1}&\color{blue}{-9}&\color{blue}{16}&\color{blue}{-64}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}-9x^{2}+16x-64 } $ with a remainder of $ \color{red}{ 0 } $.