Find remainder when $ 12x^{3}-20x^{2}+11x-14 $ is divided by $ x-2 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}2&12&-20&11&-14\\& & 24& 8& \color{black}{38} \\ \hline &\color{blue}{12}&\color{blue}{4}&\color{blue}{19}&\color{orangered}{24} \end{array} $$The remainder when $ 12x^{3}-20x^{2}+11x-14 $ is divided by $ x-2 $ is $ \, \color{red}{ 24 } $.
Explanation
We can find remainder using synthetic division method.
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.
$$ \begin{array}{c|rrrr}\color{blue}{2}&12&-20&11&-14\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}2&\color{orangered}{ 12 }&-20&11&-14\\& & & & \\ \hline &\color{orangered}{12}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 12 } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&12&-20&11&-14\\& & \color{blue}{24} & & \\ \hline &\color{blue}{12}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -20 } + \color{orangered}{ 24 } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrr}2&12&\color{orangered}{ -20 }&11&-14\\& & \color{orangered}{24} & & \\ \hline &12&\color{orangered}{4}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 4 } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&12&-20&11&-14\\& & 24& \color{blue}{8} & \\ \hline &12&\color{blue}{4}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 11 } + \color{orangered}{ 8 } = \color{orangered}{ 19 } $
$$ \begin{array}{c|rrrr}2&12&-20&\color{orangered}{ 11 }&-14\\& & 24& \color{orangered}{8} & \\ \hline &12&4&\color{orangered}{19}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 19 } = \color{blue}{ 38 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&12&-20&11&-14\\& & 24& 8& \color{blue}{38} \\ \hline &12&4&\color{blue}{19}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -14 } + \color{orangered}{ 38 } = \color{orangered}{ 24 } $
$$ \begin{array}{c|rrrr}2&12&-20&11&\color{orangered}{ -14 }\\& & 24& 8& \color{orangered}{38} \\ \hline &\color{blue}{12}&\color{blue}{4}&\color{blue}{19}&\color{orangered}{24} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 24 }\right) $.