Find remainder when $ x^{4}+5x^{3}-10x^{2}+25x $ is divided by $ x+7 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-7&1&5&-10&25&0\\& & -7& 14& -28& \color{black}{21} \\ \hline &\color{blue}{1}&\color{blue}{-2}&\color{blue}{4}&\color{blue}{-3}&\color{orangered}{21} \end{array} $$The remainder when $ x^{4}+5x^{3}-10x^{2}+25x $ is divided by $ x+7 $ is $ \, \color{red}{ 21 } $.
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 + 7 = 0 $ ( $ x = \color{blue}{ -7 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-7}&1&5&-10&25&0\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-7&\color{orangered}{ 1 }&5&-10&25&0\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ 1 } = \color{blue}{ -7 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-7}&1&5&-10&25&0\\& & \color{blue}{-7} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ \left( -7 \right) } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrr}-7&1&\color{orangered}{ 5 }&-10&25&0\\& & \color{orangered}{-7} & & & \\ \hline &1&\color{orangered}{-2}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ 14 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-7}&1&5&-10&25&0\\& & -7& \color{blue}{14} & & \\ \hline &1&\color{blue}{-2}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ 14 } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrrr}-7&1&5&\color{orangered}{ -10 }&25&0\\& & -7& \color{orangered}{14} & & \\ \hline &1&-2&\color{orangered}{4}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ 4 } = \color{blue}{ -28 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-7}&1&5&-10&25&0\\& & -7& 14& \color{blue}{-28} & \\ \hline &1&-2&\color{blue}{4}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 25 } + \color{orangered}{ \left( -28 \right) } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrrrr}-7&1&5&-10&\color{orangered}{ 25 }&0\\& & -7& 14& \color{orangered}{-28} & \\ \hline &1&-2&4&\color{orangered}{-3}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ 21 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-7}&1&5&-10&25&0\\& & -7& 14& -28& \color{blue}{21} \\ \hline &1&-2&4&\color{blue}{-3}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 21 } = \color{orangered}{ 21 } $
$$ \begin{array}{c|rrrrr}-7&1&5&-10&25&\color{orangered}{ 0 }\\& & -7& 14& -28& \color{orangered}{21} \\ \hline &\color{blue}{1}&\color{blue}{-2}&\color{blue}{4}&\color{blue}{-3}&\color{orangered}{21} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 21 }\right) $.