Divide using synthetic division $$ ( 3x^{4}-8x^{2}-11x+1 ) \div ( x-5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}5&3&0&-8&-11&1\\& & 15& 75& 335& \color{black}{1620} \\ \hline &\color{blue}{3}&\color{blue}{15}&\color{blue}{67}&\color{blue}{324}&\color{orangered}{1621} \end{array} $$The solution is:
$$ \frac{ 3x^{4}-8x^{2}-11x+1 }{ x-5 } = \color{blue}{3x^{3}+15x^{2}+67x+324} ~+~ \frac{ \color{red}{ 1621 } }{ x-5 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&3&0&-8&-11&1\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}5&\color{orangered}{ 3 }&0&-8&-11&1\\& & & & & \\ \hline &\color{orangered}{3}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&3&0&-8&-11&1\\& & \color{blue}{15} & & & \\ \hline &\color{blue}{3}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 15 } = \color{orangered}{ 15 } $
$$ \begin{array}{c|rrrrr}5&3&\color{orangered}{ 0 }&-8&-11&1\\& & \color{orangered}{15} & & & \\ \hline &3&\color{orangered}{15}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&3&0&-8&-11&1\\& & 15& \color{blue}{75} & & \\ \hline &3&\color{blue}{15}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ 75 } = \color{orangered}{ 67 } $
$$ \begin{array}{c|rrrrr}5&3&0&\color{orangered}{ -8 }&-11&1\\& & 15& \color{orangered}{75} & & \\ \hline &3&15&\color{orangered}{67}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 67 } = \color{blue}{ 335 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&3&0&-8&-11&1\\& & 15& 75& \color{blue}{335} & \\ \hline &3&15&\color{blue}{67}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -11 } + \color{orangered}{ 335 } = \color{orangered}{ 324 } $
$$ \begin{array}{c|rrrrr}5&3&0&-8&\color{orangered}{ -11 }&1\\& & 15& 75& \color{orangered}{335} & \\ \hline &3&15&67&\color{orangered}{324}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 324 } = \color{blue}{ 1620 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&3&0&-8&-11&1\\& & 15& 75& 335& \color{blue}{1620} \\ \hline &3&15&67&\color{blue}{324}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 1620 } = \color{orangered}{ 1621 } $
$$ \begin{array}{c|rrrrr}5&3&0&-8&-11&\color{orangered}{ 1 }\\& & 15& 75& 335& \color{orangered}{1620} \\ \hline &\color{blue}{3}&\color{blue}{15}&\color{blue}{67}&\color{blue}{324}&\color{orangered}{1621} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{3}+15x^{2}+67x+324 } $ with a remainder of $ \color{red}{ 1621 } $.