Find remainder when $ 9x^{3}-5x^{2}-9x $ is divided by $ 9x-5 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}5&9&-5&-9&0\\& & 45& 200& \color{black}{955} \\ \hline &\color{blue}{9}&\color{blue}{40}&\color{blue}{191}&\color{orangered}{955} \end{array} $$The remainder when $ 9x^{3}-5x^{2}-9x $ is divided by $ x-5 $ is $ \, \color{red}{ 955 } $.
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 -5 = 0 $ ( $ x = \color{blue}{ 5 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{5}&9&-5&-9&0\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}5&\color{orangered}{ 9 }&-5&-9&0\\& & & & \\ \hline &\color{orangered}{9}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 9 } = \color{blue}{ 45 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&9&-5&-9&0\\& & \color{blue}{45} & & \\ \hline &\color{blue}{9}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ 45 } = \color{orangered}{ 40 } $
$$ \begin{array}{c|rrrr}5&9&\color{orangered}{ -5 }&-9&0\\& & \color{orangered}{45} & & \\ \hline &9&\color{orangered}{40}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 40 } = \color{blue}{ 200 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&9&-5&-9&0\\& & 45& \color{blue}{200} & \\ \hline &9&\color{blue}{40}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ 200 } = \color{orangered}{ 191 } $
$$ \begin{array}{c|rrrr}5&9&-5&\color{orangered}{ -9 }&0\\& & 45& \color{orangered}{200} & \\ \hline &9&40&\color{orangered}{191}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 191 } = \color{blue}{ 955 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&9&-5&-9&0\\& & 45& 200& \color{blue}{955} \\ \hline &9&40&\color{blue}{191}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 955 } = \color{orangered}{ 955 } $
$$ \begin{array}{c|rrrr}5&9&-5&-9&\color{orangered}{ 0 }\\& & 45& 200& \color{orangered}{955} \\ \hline &\color{blue}{9}&\color{blue}{40}&\color{blue}{191}&\color{orangered}{955} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 955 }\right) $.