Divide using synthetic division $$ ( 3x^{4}-15x-9 ) \div ( x-5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}5&3&0&0&-15&-9\\& & 15& 75& 375& \color{black}{1800} \\ \hline &\color{blue}{3}&\color{blue}{15}&\color{blue}{75}&\color{blue}{360}&\color{orangered}{1791} \end{array} $$The solution is:
$$ \frac{ 3x^{4}-15x-9 }{ x-5 } = \color{blue}{3x^{3}+15x^{2}+75x+360} ~+~ \frac{ \color{red}{ 1791 } }{ x-5 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&3&0&0&-15&-9\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}5&\color{orangered}{ 3 }&0&0&-15&-9\\& & & & & \\ \hline &\color{orangered}{3}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&3&0&0&-15&-9\\& & \color{blue}{15} & & & \\ \hline &\color{blue}{3}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 15 } = \color{orangered}{ 15 } $
$$ \begin{array}{c|rrrrr}5&3&\color{orangered}{ 0 }&0&-15&-9\\& & \color{orangered}{15} & & & \\ \hline &3&\color{orangered}{15}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 15 } = \color{blue}{ 75 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&3&0&0&-15&-9\\& & 15& \color{blue}{75} & & \\ \hline &3&\color{blue}{15}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 75 } = \color{orangered}{ 75 } $
$$ \begin{array}{c|rrrrr}5&3&0&\color{orangered}{ 0 }&-15&-9\\& & 15& \color{orangered}{75} & & \\ \hline &3&15&\color{orangered}{75}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 75 } = \color{blue}{ 375 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&3&0&0&-15&-9\\& & 15& 75& \color{blue}{375} & \\ \hline &3&15&\color{blue}{75}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -15 } + \color{orangered}{ 375 } = \color{orangered}{ 360 } $
$$ \begin{array}{c|rrrrr}5&3&0&0&\color{orangered}{ -15 }&-9\\& & 15& 75& \color{orangered}{375} & \\ \hline &3&15&75&\color{orangered}{360}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 360 } = \color{blue}{ 1800 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&3&0&0&-15&-9\\& & 15& 75& 375& \color{blue}{1800} \\ \hline &3&15&75&\color{blue}{360}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ 1800 } = \color{orangered}{ 1791 } $
$$ \begin{array}{c|rrrrr}5&3&0&0&-15&\color{orangered}{ -9 }\\& & 15& 75& 375& \color{orangered}{1800} \\ \hline &\color{blue}{3}&\color{blue}{15}&\color{blue}{75}&\color{blue}{360}&\color{orangered}{1791} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{3}+15x^{2}+75x+360 } $ with a remainder of $ \color{red}{ 1791 } $.