Divide using synthetic division $$ ( 3x^{3}-15x-9 ) \div ( x-5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}5&3&0&-15&-9\\& & 15& 75& \color{black}{300} \\ \hline &\color{blue}{3}&\color{blue}{15}&\color{blue}{60}&\color{orangered}{291} \end{array} $$The solution is:
$$ \frac{ 3x^{3}-15x-9 }{ x-5 } = \color{blue}{3x^{2}+15x+60} ~+~ \frac{ \color{red}{ 291 } }{ x-5 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{5}&3&0&-15&-9\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}5&\color{orangered}{ 3 }&0&-15&-9\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&3&0&-15&-9\\& & \color{blue}{15} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 15 } = \color{orangered}{ 15 } $
$$ \begin{array}{c|rrrr}5&3&\color{orangered}{ 0 }&-15&-9\\& & \color{orangered}{15} & & \\ \hline &3&\color{orangered}{15}&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&3&0&-15&-9\\& & 15& \color{blue}{75} & \\ \hline &3&\color{blue}{15}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -15 } + \color{orangered}{ 75 } = \color{orangered}{ 60 } $
$$ \begin{array}{c|rrrr}5&3&0&\color{orangered}{ -15 }&-9\\& & 15& \color{orangered}{75} & \\ \hline &3&15&\color{orangered}{60}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 60 } = \color{blue}{ 300 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&3&0&-15&-9\\& & 15& 75& \color{blue}{300} \\ \hline &3&15&\color{blue}{60}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ 300 } = \color{orangered}{ 291 } $
$$ \begin{array}{c|rrrr}5&3&0&-15&\color{orangered}{ -9 }\\& & 15& 75& \color{orangered}{300} \\ \hline &\color{blue}{3}&\color{blue}{15}&\color{blue}{60}&\color{orangered}{291} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{2}+15x+60 } $ with a remainder of $ \color{red}{ 291 } $.