Divide using synthetic division $$ ( 2x^{3}+x^{2}-25x+21 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}3&2&1&-25&21\\& & 6& 21& \color{black}{-12} \\ \hline &\color{blue}{2}&\color{blue}{7}&\color{blue}{-4}&\color{orangered}{9} \end{array} $$The solution is:
$$ \frac{ 2x^{3}+x^{2}-25x+21 }{ x-3 } = \color{blue}{2x^{2}+7x-4} ~+~ \frac{ \color{red}{ 9 } }{ x-3 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -3 = 0 $ ( $ x = \color{blue}{ 3 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{3}&2&1&-25&21\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}3&\color{orangered}{ 2 }&1&-25&21\\& & & & \\ \hline &\color{orangered}{2}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 2 } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&2&1&-25&21\\& & \color{blue}{6} & & \\ \hline &\color{blue}{2}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 6 } = \color{orangered}{ 7 } $
$$ \begin{array}{c|rrrr}3&2&\color{orangered}{ 1 }&-25&21\\& & \color{orangered}{6} & & \\ \hline &2&\color{orangered}{7}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 7 } = \color{blue}{ 21 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&2&1&-25&21\\& & 6& \color{blue}{21} & \\ \hline &2&\color{blue}{7}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -25 } + \color{orangered}{ 21 } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrr}3&2&1&\color{orangered}{ -25 }&21\\& & 6& \color{orangered}{21} & \\ \hline &2&7&\color{orangered}{-4}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ -12 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&2&1&-25&21\\& & 6& 21& \color{blue}{-12} \\ \hline &2&7&\color{blue}{-4}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 21 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ 9 } $
$$ \begin{array}{c|rrrr}3&2&1&-25&\color{orangered}{ 21 }\\& & 6& 21& \color{orangered}{-12} \\ \hline &\color{blue}{2}&\color{blue}{7}&\color{blue}{-4}&\color{orangered}{9} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{2}+7x-4 } $ with a remainder of $ \color{red}{ 9 } $.