Divide using synthetic division $$ ( 3x^{2}-5x+9 ) \div ( x-4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}4&3&-5&9\\& & 12& \color{black}{28} \\ \hline &\color{blue}{3}&\color{blue}{7}&\color{orangered}{37} \end{array} $$The solution is:
$$ \frac{ 3x^{2}-5x+9 }{ x-4 } = \color{blue}{3x+7} ~+~ \frac{ \color{red}{ 37 } }{ x-4 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -4 = 0 $ ( $ x = \color{blue}{ 4 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{4}&3&-5&9\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}4&\color{orangered}{ 3 }&-5&9\\& & & \\ \hline &\color{orangered}{3}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 3 } = \color{blue}{ 12 } $.
$$ \begin{array}{c|rrr}\color{blue}{4}&3&-5&9\\& & \color{blue}{12} & \\ \hline &\color{blue}{3}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ 12 } = \color{orangered}{ 7 } $
$$ \begin{array}{c|rrr}4&3&\color{orangered}{ -5 }&9\\& & \color{orangered}{12} & \\ \hline &3&\color{orangered}{7}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 7 } = \color{blue}{ 28 } $.
$$ \begin{array}{c|rrr}\color{blue}{4}&3&-5&9\\& & 12& \color{blue}{28} \\ \hline &3&\color{blue}{7}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ 28 } = \color{orangered}{ 37 } $
$$ \begin{array}{c|rrr}4&3&-5&\color{orangered}{ 9 }\\& & 12& \color{orangered}{28} \\ \hline &\color{blue}{3}&\color{blue}{7}&\color{orangered}{37} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x+7 } $ with a remainder of $ \color{red}{ 37 } $.