Find remainder when $ 5x^{3}-x-25 $ is divided by $ x+5 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-5&5&0&-1&-25\\& & -25& 125& \color{black}{-620} \\ \hline &\color{blue}{5}&\color{blue}{-25}&\color{blue}{124}&\color{orangered}{-645} \end{array} $$The remainder when $ 5x^{3}-x-25 $ is divided by $ x+5 $ is $ \, \color{red}{ -645 } $.
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 + 5 = 0 $ ( $ x = \color{blue}{ -5 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&5&0&-1&-25\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-5&\color{orangered}{ 5 }&0&-1&-25\\& & & & \\ \hline &\color{orangered}{5}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 5 } = \color{blue}{ -25 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&5&0&-1&-25\\& & \color{blue}{-25} & & \\ \hline &\color{blue}{5}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -25 \right) } = \color{orangered}{ -25 } $
$$ \begin{array}{c|rrrr}-5&5&\color{orangered}{ 0 }&-1&-25\\& & \color{orangered}{-25} & & \\ \hline &5&\color{orangered}{-25}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -25 \right) } = \color{blue}{ 125 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&5&0&-1&-25\\& & -25& \color{blue}{125} & \\ \hline &5&\color{blue}{-25}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 125 } = \color{orangered}{ 124 } $
$$ \begin{array}{c|rrrr}-5&5&0&\color{orangered}{ -1 }&-25\\& & -25& \color{orangered}{125} & \\ \hline &5&-25&\color{orangered}{124}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 124 } = \color{blue}{ -620 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&5&0&-1&-25\\& & -25& 125& \color{blue}{-620} \\ \hline &5&-25&\color{blue}{124}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -25 } + \color{orangered}{ \left( -620 \right) } = \color{orangered}{ -645 } $
$$ \begin{array}{c|rrrr}-5&5&0&-1&\color{orangered}{ -25 }\\& & -25& 125& \color{orangered}{-620} \\ \hline &\color{blue}{5}&\color{blue}{-25}&\color{blue}{124}&\color{orangered}{-645} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -645 }\right) $.