Divide using synthetic division $$ ( 11x^{4}-14x-4 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}3&11&0&0&-14&-4\\& & 33& 99& 297& \color{black}{849} \\ \hline &\color{blue}{11}&\color{blue}{33}&\color{blue}{99}&\color{blue}{283}&\color{orangered}{845} \end{array} $$The solution is:
$$ \frac{ 11x^{4}-14x-4 }{ x-3 } = \color{blue}{11x^{3}+33x^{2}+99x+283} ~+~ \frac{ \color{red}{ 845 } }{ x-3 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -3 = 0 $ ( $ x = \color{blue}{ 3 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&11&0&0&-14&-4\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}3&\color{orangered}{ 11 }&0&0&-14&-4\\& & & & & \\ \hline &\color{orangered}{11}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 11 } = \color{blue}{ 33 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&11&0&0&-14&-4\\& & \color{blue}{33} & & & \\ \hline &\color{blue}{11}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 33 } = \color{orangered}{ 33 } $
$$ \begin{array}{c|rrrrr}3&11&\color{orangered}{ 0 }&0&-14&-4\\& & \color{orangered}{33} & & & \\ \hline &11&\color{orangered}{33}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 33 } = \color{blue}{ 99 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&11&0&0&-14&-4\\& & 33& \color{blue}{99} & & \\ \hline &11&\color{blue}{33}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 99 } = \color{orangered}{ 99 } $
$$ \begin{array}{c|rrrrr}3&11&0&\color{orangered}{ 0 }&-14&-4\\& & 33& \color{orangered}{99} & & \\ \hline &11&33&\color{orangered}{99}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 99 } = \color{blue}{ 297 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&11&0&0&-14&-4\\& & 33& 99& \color{blue}{297} & \\ \hline &11&33&\color{blue}{99}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -14 } + \color{orangered}{ 297 } = \color{orangered}{ 283 } $
$$ \begin{array}{c|rrrrr}3&11&0&0&\color{orangered}{ -14 }&-4\\& & 33& 99& \color{orangered}{297} & \\ \hline &11&33&99&\color{orangered}{283}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 283 } = \color{blue}{ 849 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&11&0&0&-14&-4\\& & 33& 99& 297& \color{blue}{849} \\ \hline &11&33&99&\color{blue}{283}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 849 } = \color{orangered}{ 845 } $
$$ \begin{array}{c|rrrrr}3&11&0&0&-14&\color{orangered}{ -4 }\\& & 33& 99& 297& \color{orangered}{849} \\ \hline &\color{blue}{11}&\color{blue}{33}&\color{blue}{99}&\color{blue}{283}&\color{orangered}{845} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 11x^{3}+33x^{2}+99x+283 } $ with a remainder of $ \color{red}{ 845 } $.