Find remainder when $ 9x^{3}-20x^{2}-27x+37 $ is divided by $ x-3 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}3&9&-20&-27&37\\& & 27& 21& \color{black}{-18} \\ \hline &\color{blue}{9}&\color{blue}{7}&\color{blue}{-6}&\color{orangered}{19} \end{array} $$The remainder when $ 9x^{3}-20x^{2}-27x+37 $ is divided by $ x-3 $ is $ \, \color{red}{ 19 } $.
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 -3 = 0 $ ( $ x = \color{blue}{ 3 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{3}&9&-20&-27&37\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}3&\color{orangered}{ 9 }&-20&-27&37\\& & & & \\ \hline &\color{orangered}{9}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&9&-20&-27&37\\& & \color{blue}{27} & & \\ \hline &\color{blue}{9}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -20 } + \color{orangered}{ 27 } = \color{orangered}{ 7 } $
$$ \begin{array}{c|rrrr}3&9&\color{orangered}{ -20 }&-27&37\\& & \color{orangered}{27} & & \\ \hline &9&\color{orangered}{7}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 7 } = \color{blue}{ 21 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&9&-20&-27&37\\& & 27& \color{blue}{21} & \\ \hline &9&\color{blue}{7}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -27 } + \color{orangered}{ 21 } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrr}3&9&-20&\color{orangered}{ -27 }&37\\& & 27& \color{orangered}{21} & \\ \hline &9&7&\color{orangered}{-6}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ -18 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&9&-20&-27&37\\& & 27& 21& \color{blue}{-18} \\ \hline &9&7&\color{blue}{-6}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 37 } + \color{orangered}{ \left( -18 \right) } = \color{orangered}{ 19 } $
$$ \begin{array}{c|rrrr}3&9&-20&-27&\color{orangered}{ 37 }\\& & 27& 21& \color{orangered}{-18} \\ \hline &\color{blue}{9}&\color{blue}{7}&\color{blue}{-6}&\color{orangered}{19} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 19 }\right) $.