Divide using synthetic division $$ ( 10x^{2}+24x-18 ) \div ( 5x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}3&10&24&-18\\& & 30& \color{black}{162} \\ \hline &\color{blue}{10}&\color{blue}{54}&\color{orangered}{144} \end{array} $$The solution is:
$$ \frac{ 10x^{2}+24x-18 }{ x-3 } = \color{blue}{10x+54} ~+~ \frac{ \color{red}{ 144 } }{ x-3 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{3}&10&24&-18\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}3&\color{orangered}{ 10 }&24&-18\\& & & \\ \hline &\color{orangered}{10}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 10 } = \color{blue}{ 30 } $.
$$ \begin{array}{c|rrr}\color{blue}{3}&10&24&-18\\& & \color{blue}{30} & \\ \hline &\color{blue}{10}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 24 } + \color{orangered}{ 30 } = \color{orangered}{ 54 } $
$$ \begin{array}{c|rrr}3&10&\color{orangered}{ 24 }&-18\\& & \color{orangered}{30} & \\ \hline &10&\color{orangered}{54}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 54 } = \color{blue}{ 162 } $.
$$ \begin{array}{c|rrr}\color{blue}{3}&10&24&-18\\& & 30& \color{blue}{162} \\ \hline &10&\color{blue}{54}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -18 } + \color{orangered}{ 162 } = \color{orangered}{ 144 } $
$$ \begin{array}{c|rrr}3&10&24&\color{orangered}{ -18 }\\& & 30& \color{orangered}{162} \\ \hline &\color{blue}{10}&\color{blue}{54}&\color{orangered}{144} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 10x+54 } $ with a remainder of $ \color{red}{ 144 } $.