Divide using synthetic division $$ ( 11x^{3}+14x^{2}+11x-166 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}2&11&14&11&-166\\& & 22& 72& \color{black}{166} \\ \hline &\color{blue}{11}&\color{blue}{36}&\color{blue}{83}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 11x^{3}+14x^{2}+11x-166 }{ x-2 } = \color{blue}{11x^{2}+36x+83} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -2 = 0 $ ( $ x = \color{blue}{ 2 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{2}&11&14&11&-166\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}2&\color{orangered}{ 11 }&14&11&-166\\& & & & \\ \hline &\color{orangered}{11}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 11 } = \color{blue}{ 22 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&11&14&11&-166\\& & \color{blue}{22} & & \\ \hline &\color{blue}{11}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 14 } + \color{orangered}{ 22 } = \color{orangered}{ 36 } $
$$ \begin{array}{c|rrrr}2&11&\color{orangered}{ 14 }&11&-166\\& & \color{orangered}{22} & & \\ \hline &11&\color{orangered}{36}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 36 } = \color{blue}{ 72 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&11&14&11&-166\\& & 22& \color{blue}{72} & \\ \hline &11&\color{blue}{36}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 11 } + \color{orangered}{ 72 } = \color{orangered}{ 83 } $
$$ \begin{array}{c|rrrr}2&11&14&\color{orangered}{ 11 }&-166\\& & 22& \color{orangered}{72} & \\ \hline &11&36&\color{orangered}{83}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 83 } = \color{blue}{ 166 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&11&14&11&-166\\& & 22& 72& \color{blue}{166} \\ \hline &11&36&\color{blue}{83}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -166 } + \color{orangered}{ 166 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}2&11&14&11&\color{orangered}{ -166 }\\& & 22& 72& \color{orangered}{166} \\ \hline &\color{blue}{11}&\color{blue}{36}&\color{blue}{83}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 11x^{2}+36x+83 } $ with a remainder of $ \color{red}{ 0 } $.