Find remainder when $ x^{2}-2x-24 $ is divided by $ x+4 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-4&1&-2&-24\\& & -4& \color{black}{24} \\ \hline &\color{blue}{1}&\color{blue}{-6}&\color{orangered}{0} \end{array} $$The remainder when $ x^{2}-2x-24 $ is divided by $ x+4 $ is $ \, \color{red}{ 0 } $.
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 + 4 = 0 $ ( $ x = \color{blue}{ -4 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{-4}&1&-2&-24\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-4&\color{orangered}{ 1 }&-2&-24\\& & & \\ \hline &\color{orangered}{1}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 1 } = \color{blue}{ -4 } $.
$$ \begin{array}{c|rrr}\color{blue}{-4}&1&-2&-24\\& & \color{blue}{-4} & \\ \hline &\color{blue}{1}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrr}-4&1&\color{orangered}{ -2 }&-24\\& & \color{orangered}{-4} & \\ \hline &1&\color{orangered}{-6}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrr}\color{blue}{-4}&1&-2&-24\\& & -4& \color{blue}{24} \\ \hline &1&\color{blue}{-6}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -24 } + \color{orangered}{ 24 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrr}-4&1&-2&\color{orangered}{ -24 }\\& & -4& \color{orangered}{24} \\ \hline &\color{blue}{1}&\color{blue}{-6}&\color{orangered}{0} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 0 }\right) $.