Divide using synthetic division $$ ( 3x^{2}+5x-2 ) \div ( x-5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}5&3&5&-2\\& & 15& \color{black}{100} \\ \hline &\color{blue}{3}&\color{blue}{20}&\color{orangered}{98} \end{array} $$The solution is:
$$ \frac{ 3x^{2}+5x-2 }{ x-5 } = \color{blue}{3x+20} ~+~ \frac{ \color{red}{ 98 } }{ x-5 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{5}&3&5&-2\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}5&\color{orangered}{ 3 }&5&-2\\& & & \\ \hline &\color{orangered}{3}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 3 } = \color{blue}{ 15 } $.
$$ \begin{array}{c|rrr}\color{blue}{5}&3&5&-2\\& & \color{blue}{15} & \\ \hline &\color{blue}{3}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 15 } = \color{orangered}{ 20 } $
$$ \begin{array}{c|rrr}5&3&\color{orangered}{ 5 }&-2\\& & \color{orangered}{15} & \\ \hline &3&\color{orangered}{20}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 20 } = \color{blue}{ 100 } $.
$$ \begin{array}{c|rrr}\color{blue}{5}&3&5&-2\\& & 15& \color{blue}{100} \\ \hline &3&\color{blue}{20}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 100 } = \color{orangered}{ 98 } $
$$ \begin{array}{c|rrr}5&3&5&\color{orangered}{ -2 }\\& & 15& \color{orangered}{100} \\ \hline &\color{blue}{3}&\color{blue}{20}&\color{orangered}{98} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x+20 } $ with a remainder of $ \color{red}{ 98 } $.