Find remainder when $ 2x^{3}-3x^{2}-50x+75 $ is divided by $ x-1 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}1&2&-3&-50&75\\& & 2& -1& \color{black}{-51} \\ \hline &\color{blue}{2}&\color{blue}{-1}&\color{blue}{-51}&\color{orangered}{24} \end{array} $$The remainder when $ 2x^{3}-3x^{2}-50x+75 $ is divided by $ x-1 $ is $ \, \color{red}{ 24 } $.
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 -1 = 0 $ ( $ x = \color{blue}{ 1 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{1}&2&-3&-50&75\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}1&\color{orangered}{ 2 }&-3&-50&75\\& & & & \\ \hline &\color{orangered}{2}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 2 } = \color{blue}{ 2 } $.
$$ \begin{array}{c|rrrr}\color{blue}{1}&2&-3&-50&75\\& & \color{blue}{2} & & \\ \hline &\color{blue}{2}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 2 } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrr}1&2&\color{orangered}{ -3 }&-50&75\\& & \color{orangered}{2} & & \\ \hline &2&\color{orangered}{-1}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ -1 } $.
$$ \begin{array}{c|rrrr}\color{blue}{1}&2&-3&-50&75\\& & 2& \color{blue}{-1} & \\ \hline &2&\color{blue}{-1}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -50 } + \color{orangered}{ \left( -1 \right) } = \color{orangered}{ -51 } $
$$ \begin{array}{c|rrrr}1&2&-3&\color{orangered}{ -50 }&75\\& & 2& \color{orangered}{-1} & \\ \hline &2&-1&\color{orangered}{-51}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \left( -51 \right) } = \color{blue}{ -51 } $.
$$ \begin{array}{c|rrrr}\color{blue}{1}&2&-3&-50&75\\& & 2& -1& \color{blue}{-51} \\ \hline &2&-1&\color{blue}{-51}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 75 } + \color{orangered}{ \left( -51 \right) } = \color{orangered}{ 24 } $
$$ \begin{array}{c|rrrr}1&2&-3&-50&\color{orangered}{ 75 }\\& & 2& -1& \color{orangered}{-51} \\ \hline &\color{blue}{2}&\color{blue}{-1}&\color{blue}{-51}&\color{orangered}{24} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 24 }\right) $.