Divide using synthetic division $$ ( 2x^{4}+7x^{3}-24x^{2} ) \div ( 27x-18 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}18&2&7&-24&0&0\\& & 36& 774& 13500& \color{black}{243000} \\ \hline &\color{blue}{2}&\color{blue}{43}&\color{blue}{750}&\color{blue}{13500}&\color{orangered}{243000} \end{array} $$The solution is:
$$ \frac{ 2x^{4}+7x^{3}-24x^{2} }{ x-18 } = \color{blue}{2x^{3}+43x^{2}+750x+13500} ~+~ \frac{ \color{red}{ 243000 } }{ x-18 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -18 = 0 $ ( $ x = \color{blue}{ 18 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{18}&2&7&-24&0&0\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}18&\color{orangered}{ 2 }&7&-24&0&0\\& & & & & \\ \hline &\color{orangered}{2}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 18 } \cdot \color{blue}{ 2 } = \color{blue}{ 36 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{18}&2&7&-24&0&0\\& & \color{blue}{36} & & & \\ \hline &\color{blue}{2}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 7 } + \color{orangered}{ 36 } = \color{orangered}{ 43 } $
$$ \begin{array}{c|rrrrr}18&2&\color{orangered}{ 7 }&-24&0&0\\& & \color{orangered}{36} & & & \\ \hline &2&\color{orangered}{43}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 18 } \cdot \color{blue}{ 43 } = \color{blue}{ 774 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{18}&2&7&-24&0&0\\& & 36& \color{blue}{774} & & \\ \hline &2&\color{blue}{43}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -24 } + \color{orangered}{ 774 } = \color{orangered}{ 750 } $
$$ \begin{array}{c|rrrrr}18&2&7&\color{orangered}{ -24 }&0&0\\& & 36& \color{orangered}{774} & & \\ \hline &2&43&\color{orangered}{750}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 18 } \cdot \color{blue}{ 750 } = \color{blue}{ 13500 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{18}&2&7&-24&0&0\\& & 36& 774& \color{blue}{13500} & \\ \hline &2&43&\color{blue}{750}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 13500 } = \color{orangered}{ 13500 } $
$$ \begin{array}{c|rrrrr}18&2&7&-24&\color{orangered}{ 0 }&0\\& & 36& 774& \color{orangered}{13500} & \\ \hline &2&43&750&\color{orangered}{13500}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 18 } \cdot \color{blue}{ 13500 } = \color{blue}{ 243000 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{18}&2&7&-24&0&0\\& & 36& 774& 13500& \color{blue}{243000} \\ \hline &2&43&750&\color{blue}{13500}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 243000 } = \color{orangered}{ 243000 } $
$$ \begin{array}{c|rrrrr}18&2&7&-24&0&\color{orangered}{ 0 }\\& & 36& 774& 13500& \color{orangered}{243000} \\ \hline &\color{blue}{2}&\color{blue}{43}&\color{blue}{750}&\color{blue}{13500}&\color{orangered}{243000} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{3}+43x^{2}+750x+13500 } $ with a remainder of $ \color{red}{ 243000 } $.