Find remainder when $ x^{3}-4x^{2}-23x-32 $ is divided by $ x-2 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}2&1&-4&-23&-32\\& & 2& -4& \color{black}{-54} \\ \hline &\color{blue}{1}&\color{blue}{-2}&\color{blue}{-27}&\color{orangered}{-86} \end{array} $$The remainder when $ x^{3}-4x^{2}-23x-32 $ is divided by $ x-2 $ is $ \, \color{red}{ -86 } $.
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.
$$ \begin{array}{c|rrrr}\color{blue}{2}&1&-4&-23&-32\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}2&\color{orangered}{ 1 }&-4&-23&-32\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 1 } = \color{blue}{ 2 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&1&-4&-23&-32\\& & \color{blue}{2} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 2 } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrr}2&1&\color{orangered}{ -4 }&-23&-32\\& & \color{orangered}{2} & & \\ \hline &1&\color{orangered}{-2}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ -4 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&1&-4&-23&-32\\& & 2& \color{blue}{-4} & \\ \hline &1&\color{blue}{-2}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -23 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ -27 } $
$$ \begin{array}{c|rrrr}2&1&-4&\color{orangered}{ -23 }&-32\\& & 2& \color{orangered}{-4} & \\ \hline &1&-2&\color{orangered}{-27}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -27 \right) } = \color{blue}{ -54 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&1&-4&-23&-32\\& & 2& -4& \color{blue}{-54} \\ \hline &1&-2&\color{blue}{-27}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -32 } + \color{orangered}{ \left( -54 \right) } = \color{orangered}{ -86 } $
$$ \begin{array}{c|rrrr}2&1&-4&-23&\color{orangered}{ -32 }\\& & 2& -4& \color{orangered}{-54} \\ \hline &\color{blue}{1}&\color{blue}{-2}&\color{blue}{-27}&\color{orangered}{-86} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -86 }\right) $.