Find remainder when $ 2x^{3}-3x^{2}-50x+75 $ is divided by $ x-3 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}3&2&-3&-50&75\\& & 6& 9& \color{black}{-123} \\ \hline &\color{blue}{2}&\color{blue}{3}&\color{blue}{-41}&\color{orangered}{-48} \end{array} $$The remainder when $ 2x^{3}-3x^{2}-50x+75 $ is divided by $ x-3 $ is $ \, \color{red}{ -48 } $.
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.
$$ \begin{array}{c|rrrr}\color{blue}{3}&2&-3&-50&75\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}3&\color{orangered}{ 2 }&-3&-50&75\\& & & & \\ \hline &\color{orangered}{2}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 2 } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&2&-3&-50&75\\& & \color{blue}{6} & & \\ \hline &\color{blue}{2}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 6 } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrr}3&2&\color{orangered}{ -3 }&-50&75\\& & \color{orangered}{6} & & \\ \hline &2&\color{orangered}{3}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 3 } = \color{blue}{ 9 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&2&-3&-50&75\\& & 6& \color{blue}{9} & \\ \hline &2&\color{blue}{3}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -50 } + \color{orangered}{ 9 } = \color{orangered}{ -41 } $
$$ \begin{array}{c|rrrr}3&2&-3&\color{orangered}{ -50 }&75\\& & 6& \color{orangered}{9} & \\ \hline &2&3&\color{orangered}{-41}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -41 \right) } = \color{blue}{ -123 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&2&-3&-50&75\\& & 6& 9& \color{blue}{-123} \\ \hline &2&3&\color{blue}{-41}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 75 } + \color{orangered}{ \left( -123 \right) } = \color{orangered}{ -48 } $
$$ \begin{array}{c|rrrr}3&2&-3&-50&\color{orangered}{ 75 }\\& & 6& 9& \color{orangered}{-123} \\ \hline &\color{blue}{2}&\color{blue}{3}&\color{blue}{-41}&\color{orangered}{-48} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -48 }\right) $.