Divide using synthetic division $$ ( 12x^{2}+x-35 ) \div ( 3x-5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}5&12&1&-35\\& & 60& \color{black}{305} \\ \hline &\color{blue}{12}&\color{blue}{61}&\color{orangered}{270} \end{array} $$The solution is:
$$ \frac{ 12x^{2}+x-35 }{ x-5 } = \color{blue}{12x+61} ~+~ \frac{ \color{red}{ 270 } }{ x-5 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -5 = 0 $ ( $ x = \color{blue}{ 5 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{5}&12&1&-35\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}5&\color{orangered}{ 12 }&1&-35\\& & & \\ \hline &\color{orangered}{12}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 12 } = \color{blue}{ 60 } $.
$$ \begin{array}{c|rrr}\color{blue}{5}&12&1&-35\\& & \color{blue}{60} & \\ \hline &\color{blue}{12}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 60 } = \color{orangered}{ 61 } $
$$ \begin{array}{c|rrr}5&12&\color{orangered}{ 1 }&-35\\& & \color{orangered}{60} & \\ \hline &12&\color{orangered}{61}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 61 } = \color{blue}{ 305 } $.
$$ \begin{array}{c|rrr}\color{blue}{5}&12&1&-35\\& & 60& \color{blue}{305} \\ \hline &12&\color{blue}{61}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -35 } + \color{orangered}{ 305 } = \color{orangered}{ 270 } $
$$ \begin{array}{c|rrr}5&12&1&\color{orangered}{ -35 }\\& & 60& \color{orangered}{305} \\ \hline &\color{blue}{12}&\color{blue}{61}&\color{orangered}{270} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 12x+61 } $ with a remainder of $ \color{red}{ 270 } $.