Find remainder when $ x^{3}-2x^{2}-30x-35 $ is divided by $ x-9 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}9&1&-2&-30&-35\\& & 9& 63& \color{black}{297} \\ \hline &\color{blue}{1}&\color{blue}{7}&\color{blue}{33}&\color{orangered}{262} \end{array} $$The remainder when $ x^{3}-2x^{2}-30x-35 $ is divided by $ x-9 $ is $ \, \color{red}{ 262 } $.
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 -9 = 0 $ ( $ x = \color{blue}{ 9 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{9}&1&-2&-30&-35\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}9&\color{orangered}{ 1 }&-2&-30&-35\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ 1 } = \color{blue}{ 9 } $.
$$ \begin{array}{c|rrrr}\color{blue}{9}&1&-2&-30&-35\\& & \color{blue}{9} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 9 } = \color{orangered}{ 7 } $
$$ \begin{array}{c|rrrr}9&1&\color{orangered}{ -2 }&-30&-35\\& & \color{orangered}{9} & & \\ \hline &1&\color{orangered}{7}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ 7 } = \color{blue}{ 63 } $.
$$ \begin{array}{c|rrrr}\color{blue}{9}&1&-2&-30&-35\\& & 9& \color{blue}{63} & \\ \hline &1&\color{blue}{7}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -30 } + \color{orangered}{ 63 } = \color{orangered}{ 33 } $
$$ \begin{array}{c|rrrr}9&1&-2&\color{orangered}{ -30 }&-35\\& & 9& \color{orangered}{63} & \\ \hline &1&7&\color{orangered}{33}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ 33 } = \color{blue}{ 297 } $.
$$ \begin{array}{c|rrrr}\color{blue}{9}&1&-2&-30&-35\\& & 9& 63& \color{blue}{297} \\ \hline &1&7&\color{blue}{33}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -35 } + \color{orangered}{ 297 } = \color{orangered}{ 262 } $
$$ \begin{array}{c|rrrr}9&1&-2&-30&\color{orangered}{ -35 }\\& & 9& 63& \color{orangered}{297} \\ \hline &\color{blue}{1}&\color{blue}{7}&\color{blue}{33}&\color{orangered}{262} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 262 }\right) $.