Divide using synthetic division $$ ( 9x^{2}-34x+21 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}3&9&-34&21\\& & 27& \color{black}{-21} \\ \hline &\color{blue}{9}&\color{blue}{-7}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 9x^{2}-34x+21 }{ x-3 } = \color{blue}{9x-7} $$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.
$$ \begin{array}{c|rrr}\color{blue}{3}&9&-34&21\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}3&\color{orangered}{ 9 }&-34&21\\& & & \\ \hline &\color{orangered}{9}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 9 } = \color{blue}{ 27 } $.
$$ \begin{array}{c|rrr}\color{blue}{3}&9&-34&21\\& & \color{blue}{27} & \\ \hline &\color{blue}{9}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -34 } + \color{orangered}{ 27 } = \color{orangered}{ -7 } $
$$ \begin{array}{c|rrr}3&9&\color{orangered}{ -34 }&21\\& & \color{orangered}{27} & \\ \hline &9&\color{orangered}{-7}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -7 \right) } = \color{blue}{ -21 } $.
$$ \begin{array}{c|rrr}\color{blue}{3}&9&-34&21\\& & 27& \color{blue}{-21} \\ \hline &9&\color{blue}{-7}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 21 } + \color{orangered}{ \left( -21 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrr}3&9&-34&\color{orangered}{ 21 }\\& & 27& \color{orangered}{-21} \\ \hline &\color{blue}{9}&\color{blue}{-7}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 9x-7 } $ with a remainder of $ \color{red}{ 0 } $.