Find remainder when $ x^{3}+2x^{2}-4x+1 $ is divided by $ x+5 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-5&1&2&-4&1\\& & -5& 15& \color{black}{-55} \\ \hline &\color{blue}{1}&\color{blue}{-3}&\color{blue}{11}&\color{orangered}{-54} \end{array} $$The remainder when $ x^{3}+2x^{2}-4x+1 $ is divided by $ x+5 $ is $ \, \color{red}{ -54 } $.
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.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&1&2&-4&1\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-5&\color{orangered}{ 1 }&2&-4&1\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 1 } = \color{blue}{ -5 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&1&2&-4&1\\& & \color{blue}{-5} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -5 \right) } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrrr}-5&1&\color{orangered}{ 2 }&-4&1\\& & \color{orangered}{-5} & & \\ \hline &1&\color{orangered}{-3}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ 15 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&1&2&-4&1\\& & -5& \color{blue}{15} & \\ \hline &1&\color{blue}{-3}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 15 } = \color{orangered}{ 11 } $
$$ \begin{array}{c|rrrr}-5&1&2&\color{orangered}{ -4 }&1\\& & -5& \color{orangered}{15} & \\ \hline &1&-3&\color{orangered}{11}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 11 } = \color{blue}{ -55 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&1&2&-4&1\\& & -5& 15& \color{blue}{-55} \\ \hline &1&-3&\color{blue}{11}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ \left( -55 \right) } = \color{orangered}{ -54 } $
$$ \begin{array}{c|rrrr}-5&1&2&-4&\color{orangered}{ 1 }\\& & -5& 15& \color{orangered}{-55} \\ \hline &\color{blue}{1}&\color{blue}{-3}&\color{blue}{11}&\color{orangered}{-54} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -54 }\right) $.