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