Find remainder when $ 4x^{3}+x^{2}-5x-6 $ is divided by $ x-2 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}2&4&1&-5&-6\\& & 8& 18& \color{black}{26} \\ \hline &\color{blue}{4}&\color{blue}{9}&\color{blue}{13}&\color{orangered}{20} \end{array} $$The remainder when $ 4x^{3}+x^{2}-5x-6 $ is divided by $ x-2 $ is $ \, \color{red}{ 20 } $.
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.
$$ \begin{array}{c|rrrr}\color{blue}{2}&4&1&-5&-6\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}2&\color{orangered}{ 4 }&1&-5&-6\\& & & & \\ \hline &\color{orangered}{4}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 4 } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&4&1&-5&-6\\& & \color{blue}{8} & & \\ \hline &\color{blue}{4}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 8 } = \color{orangered}{ 9 } $
$$ \begin{array}{c|rrrr}2&4&\color{orangered}{ 1 }&-5&-6\\& & \color{orangered}{8} & & \\ \hline &4&\color{orangered}{9}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 9 } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&4&1&-5&-6\\& & 8& \color{blue}{18} & \\ \hline &4&\color{blue}{9}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ 18 } = \color{orangered}{ 13 } $
$$ \begin{array}{c|rrrr}2&4&1&\color{orangered}{ -5 }&-6\\& & 8& \color{orangered}{18} & \\ \hline &4&9&\color{orangered}{13}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 13 } = \color{blue}{ 26 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&4&1&-5&-6\\& & 8& 18& \color{blue}{26} \\ \hline &4&9&\color{blue}{13}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ 26 } = \color{orangered}{ 20 } $
$$ \begin{array}{c|rrrr}2&4&1&-5&\color{orangered}{ -6 }\\& & 8& 18& \color{orangered}{26} \\ \hline &\color{blue}{4}&\color{blue}{9}&\color{blue}{13}&\color{orangered}{20} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 20 }\right) $.