Divide using synthetic division $$ ( 12x^{3}+2x^{2}-5x+1 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}3&12&2&-5&1\\& & 36& 114& \color{black}{327} \\ \hline &\color{blue}{12}&\color{blue}{38}&\color{blue}{109}&\color{orangered}{328} \end{array} $$The solution is:
$$ \frac{ 12x^{3}+2x^{2}-5x+1 }{ x-3 } = \color{blue}{12x^{2}+38x+109} ~+~ \frac{ \color{red}{ 328 } }{ x-3 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{3}&12&2&-5&1\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}3&\color{orangered}{ 12 }&2&-5&1\\& & & & \\ \hline &\color{orangered}{12}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 12 } = \color{blue}{ 36 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&12&2&-5&1\\& & \color{blue}{36} & & \\ \hline &\color{blue}{12}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 36 } = \color{orangered}{ 38 } $
$$ \begin{array}{c|rrrr}3&12&\color{orangered}{ 2 }&-5&1\\& & \color{orangered}{36} & & \\ \hline &12&\color{orangered}{38}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 38 } = \color{blue}{ 114 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&12&2&-5&1\\& & 36& \color{blue}{114} & \\ \hline &12&\color{blue}{38}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ 114 } = \color{orangered}{ 109 } $
$$ \begin{array}{c|rrrr}3&12&2&\color{orangered}{ -5 }&1\\& & 36& \color{orangered}{114} & \\ \hline &12&38&\color{orangered}{109}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 109 } = \color{blue}{ 327 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&12&2&-5&1\\& & 36& 114& \color{blue}{327} \\ \hline &12&38&\color{blue}{109}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 327 } = \color{orangered}{ 328 } $
$$ \begin{array}{c|rrrr}3&12&2&-5&\color{orangered}{ 1 }\\& & 36& 114& \color{orangered}{327} \\ \hline &\color{blue}{12}&\color{blue}{38}&\color{blue}{109}&\color{orangered}{328} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 12x^{2}+38x+109 } $ with a remainder of $ \color{red}{ 328 } $.