Find remainder when $ -x^{4}+21x^{2}+18x+10 $ is divided by $ x-5 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}5&-1&0&21&18&10\\& & -5& -25& -20& \color{black}{-10} \\ \hline &\color{blue}{-1}&\color{blue}{-5}&\color{blue}{-4}&\color{blue}{-2}&\color{orangered}{0} \end{array} $$The remainder when $ -x^{4}+21x^{2}+18x+10 $ is divided by $ x-5 $ 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 -5 = 0 $ ( $ x = \color{blue}{ 5 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&-1&0&21&18&10\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}5&\color{orangered}{ -1 }&0&21&18&10\\& & & & & \\ \hline &\color{orangered}{-1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ -5 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&-1&0&21&18&10\\& & \color{blue}{-5} & & & \\ \hline &\color{blue}{-1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -5 \right) } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrrrr}5&-1&\color{orangered}{ 0 }&21&18&10\\& & \color{orangered}{-5} & & & \\ \hline &-1&\color{orangered}{-5}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ -25 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&-1&0&21&18&10\\& & -5& \color{blue}{-25} & & \\ \hline &-1&\color{blue}{-5}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 21 } + \color{orangered}{ \left( -25 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrrr}5&-1&0&\color{orangered}{ 21 }&18&10\\& & -5& \color{orangered}{-25} & & \\ \hline &-1&-5&\color{orangered}{-4}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ -20 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&-1&0&21&18&10\\& & -5& -25& \color{blue}{-20} & \\ \hline &-1&-5&\color{blue}{-4}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 18 } + \color{orangered}{ \left( -20 \right) } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrr}5&-1&0&21&\color{orangered}{ 18 }&10\\& & -5& -25& \color{orangered}{-20} & \\ \hline &-1&-5&-4&\color{orangered}{-2}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ -10 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&-1&0&21&18&10\\& & -5& -25& -20& \color{blue}{-10} \\ \hline &-1&-5&-4&\color{blue}{-2}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 10 } + \color{orangered}{ \left( -10 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}5&-1&0&21&18&\color{orangered}{ 10 }\\& & -5& -25& -20& \color{orangered}{-10} \\ \hline &\color{blue}{-1}&\color{blue}{-5}&\color{blue}{-4}&\color{blue}{-2}&\color{orangered}{0} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 0 }\right) $.