Divide using synthetic division $$ ( 48x^{3}-80x^{2}+41x-6 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}2&48&-80&41&-6\\& & 96& 32& \color{black}{146} \\ \hline &\color{blue}{48}&\color{blue}{16}&\color{blue}{73}&\color{orangered}{140} \end{array} $$The solution is:
$$ \frac{ 48x^{3}-80x^{2}+41x-6 }{ x-2 } = \color{blue}{48x^{2}+16x+73} ~+~ \frac{ \color{red}{ 140 } }{ x-2 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -2 = 0 $ ( $ x = \color{blue}{ 2 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{2}&48&-80&41&-6\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}2&\color{orangered}{ 48 }&-80&41&-6\\& & & & \\ \hline &\color{orangered}{48}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 48 } = \color{blue}{ 96 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&48&-80&41&-6\\& & \color{blue}{96} & & \\ \hline &\color{blue}{48}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -80 } + \color{orangered}{ 96 } = \color{orangered}{ 16 } $
$$ \begin{array}{c|rrrr}2&48&\color{orangered}{ -80 }&41&-6\\& & \color{orangered}{96} & & \\ \hline &48&\color{orangered}{16}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 16 } = \color{blue}{ 32 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&48&-80&41&-6\\& & 96& \color{blue}{32} & \\ \hline &48&\color{blue}{16}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 41 } + \color{orangered}{ 32 } = \color{orangered}{ 73 } $
$$ \begin{array}{c|rrrr}2&48&-80&\color{orangered}{ 41 }&-6\\& & 96& \color{orangered}{32} & \\ \hline &48&16&\color{orangered}{73}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 73 } = \color{blue}{ 146 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&48&-80&41&-6\\& & 96& 32& \color{blue}{146} \\ \hline &48&16&\color{blue}{73}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ 146 } = \color{orangered}{ 140 } $
$$ \begin{array}{c|rrrr}2&48&-80&41&\color{orangered}{ -6 }\\& & 96& 32& \color{orangered}{146} \\ \hline &\color{blue}{48}&\color{blue}{16}&\color{blue}{73}&\color{orangered}{140} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 48x^{2}+16x+73 } $ with a remainder of $ \color{red}{ 140 } $.