Divide using synthetic division $$ ( 2x^{4}-21x^{3}+8x^{2}+18x+20 ) \div ( x-10 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}10&2&-21&8&18&20\\& & 20& -10& -20& \color{black}{-20} \\ \hline &\color{blue}{2}&\color{blue}{-1}&\color{blue}{-2}&\color{blue}{-2}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 2x^{4}-21x^{3}+8x^{2}+18x+20 }{ x-10 } = \color{blue}{2x^{3}-x^{2}-2x-2} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -10 = 0 $ ( $ x = \color{blue}{ 10 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{10}&2&-21&8&18&20\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}10&\color{orangered}{ 2 }&-21&8&18&20\\& & & & & \\ \hline &\color{orangered}{2}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 10 } \cdot \color{blue}{ 2 } = \color{blue}{ 20 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{10}&2&-21&8&18&20\\& & \color{blue}{20} & & & \\ \hline &\color{blue}{2}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -21 } + \color{orangered}{ 20 } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrrr}10&2&\color{orangered}{ -21 }&8&18&20\\& & \color{orangered}{20} & & & \\ \hline &2&\color{orangered}{-1}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 10 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ -10 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{10}&2&-21&8&18&20\\& & 20& \color{blue}{-10} & & \\ \hline &2&\color{blue}{-1}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ \left( -10 \right) } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrr}10&2&-21&\color{orangered}{ 8 }&18&20\\& & 20& \color{orangered}{-10} & & \\ \hline &2&-1&\color{orangered}{-2}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 10 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ -20 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{10}&2&-21&8&18&20\\& & 20& -10& \color{blue}{-20} & \\ \hline &2&-1&\color{blue}{-2}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 18 } + \color{orangered}{ \left( -20 \right) } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrr}10&2&-21&8&\color{orangered}{ 18 }&20\\& & 20& -10& \color{orangered}{-20} & \\ \hline &2&-1&-2&\color{orangered}{-2}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 10 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ -20 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{10}&2&-21&8&18&20\\& & 20& -10& -20& \color{blue}{-20} \\ \hline &2&-1&-2&\color{blue}{-2}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 20 } + \color{orangered}{ \left( -20 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}10&2&-21&8&18&\color{orangered}{ 20 }\\& & 20& -10& -20& \color{orangered}{-20} \\ \hline &\color{blue}{2}&\color{blue}{-1}&\color{blue}{-2}&\color{blue}{-2}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{3}-x^{2}-2x-2 } $ with a remainder of $ \color{red}{ 0 } $.