Find remainder when $ 9x^{2}-30x-15 $ is divided by $ x-4 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}4&9&-30&-15\\& & 36& \color{black}{24} \\ \hline &\color{blue}{9}&\color{blue}{6}&\color{orangered}{9} \end{array} $$The remainder when $ 9x^{2}-30x-15 $ is divided by $ x-4 $ is $ \, \color{red}{ 9 } $.
Explanation
We can find remainder using synthetic division method.
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.
$$ \begin{array}{c|rrr}\color{blue}{4}&9&-30&-15\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}4&\color{orangered}{ 9 }&-30&-15\\& & & \\ \hline &\color{orangered}{9}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 9 } = \color{blue}{ 36 } $.
$$ \begin{array}{c|rrr}\color{blue}{4}&9&-30&-15\\& & \color{blue}{36} & \\ \hline &\color{blue}{9}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -30 } + \color{orangered}{ 36 } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrr}4&9&\color{orangered}{ -30 }&-15\\& & \color{orangered}{36} & \\ \hline &9&\color{orangered}{6}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 6 } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrr}\color{blue}{4}&9&-30&-15\\& & 36& \color{blue}{24} \\ \hline &9&\color{blue}{6}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -15 } + \color{orangered}{ 24 } = \color{orangered}{ 9 } $
$$ \begin{array}{c|rrr}4&9&-30&\color{orangered}{ -15 }\\& & 36& \color{orangered}{24} \\ \hline &\color{blue}{9}&\color{blue}{6}&\color{orangered}{9} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 9 }\right) $.