Divide using synthetic division $$ ( 3x^{5}+8x^{3}-5x-2 ) \div ( x-5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}5&3&0&8&0&-5&-2\\& & 15& 75& 415& 2075& \color{black}{10350} \\ \hline &\color{blue}{3}&\color{blue}{15}&\color{blue}{83}&\color{blue}{415}&\color{blue}{2070}&\color{orangered}{10348} \end{array} $$The solution is:
$$ \frac{ 3x^{5}+8x^{3}-5x-2 }{ x-5 } = \color{blue}{3x^{4}+15x^{3}+83x^{2}+415x+2070} ~+~ \frac{ \color{red}{ 10348 } }{ x-5 } $$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.
$$ \begin{array}{c|rrrrrr}\color{blue}{5}&3&0&8&0&-5&-2\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}5&\color{orangered}{ 3 }&0&8&0&-5&-2\\& & & & & & \\ \hline &\color{orangered}{3}&&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{5}&3&0&8&0&-5&-2\\& & \color{blue}{15} & & & & \\ \hline &\color{blue}{3}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 15 } = \color{orangered}{ 15 } $
$$ \begin{array}{c|rrrrrr}5&3&\color{orangered}{ 0 }&8&0&-5&-2\\& & \color{orangered}{15} & & & & \\ \hline &3&\color{orangered}{15}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{5}&3&0&8&0&-5&-2\\& & 15& \color{blue}{75} & & & \\ \hline &3&\color{blue}{15}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ 75 } = \color{orangered}{ 83 } $
$$ \begin{array}{c|rrrrrr}5&3&0&\color{orangered}{ 8 }&0&-5&-2\\& & 15& \color{orangered}{75} & & & \\ \hline &3&15&\color{orangered}{83}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 83 } = \color{blue}{ 415 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{5}&3&0&8&0&-5&-2\\& & 15& 75& \color{blue}{415} & & \\ \hline &3&15&\color{blue}{83}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 415 } = \color{orangered}{ 415 } $
$$ \begin{array}{c|rrrrrr}5&3&0&8&\color{orangered}{ 0 }&-5&-2\\& & 15& 75& \color{orangered}{415} & & \\ \hline &3&15&83&\color{orangered}{415}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 415 } = \color{blue}{ 2075 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{5}&3&0&8&0&-5&-2\\& & 15& 75& 415& \color{blue}{2075} & \\ \hline &3&15&83&\color{blue}{415}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ 2075 } = \color{orangered}{ 2070 } $
$$ \begin{array}{c|rrrrrr}5&3&0&8&0&\color{orangered}{ -5 }&-2\\& & 15& 75& 415& \color{orangered}{2075} & \\ \hline &3&15&83&415&\color{orangered}{2070}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 2070 } = \color{blue}{ 10350 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{5}&3&0&8&0&-5&-2\\& & 15& 75& 415& 2075& \color{blue}{10350} \\ \hline &3&15&83&415&\color{blue}{2070}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 10350 } = \color{orangered}{ 10348 } $
$$ \begin{array}{c|rrrrrr}5&3&0&8&0&-5&\color{orangered}{ -2 }\\& & 15& 75& 415& 2075& \color{orangered}{10350} \\ \hline &\color{blue}{3}&\color{blue}{15}&\color{blue}{83}&\color{blue}{415}&\color{blue}{2070}&\color{orangered}{10348} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{4}+15x^{3}+83x^{2}+415x+2070 } $ with a remainder of $ \color{red}{ 10348 } $.