Find remainder when $ 2x^{3}-3x^{2}-50x+75 $ is divided by $ x-5 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}5&2&-3&-50&75\\& & 10& 35& \color{black}{-75} \\ \hline &\color{blue}{2}&\color{blue}{7}&\color{blue}{-15}&\color{orangered}{0} \end{array} $$The remainder when $ 2x^{3}-3x^{2}-50x+75 $ is divided by $ x-5 $ is $ \, \color{red}{ 0 } $.
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.
$$ \begin{array}{c|rrrr}\color{blue}{5}&2&-3&-50&75\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}5&\color{orangered}{ 2 }&-3&-50&75\\& & & & \\ \hline &\color{orangered}{2}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 2 } = \color{blue}{ 10 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&2&-3&-50&75\\& & \color{blue}{10} & & \\ \hline &\color{blue}{2}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 10 } = \color{orangered}{ 7 } $
$$ \begin{array}{c|rrrr}5&2&\color{orangered}{ -3 }&-50&75\\& & \color{orangered}{10} & & \\ \hline &2&\color{orangered}{7}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 7 } = \color{blue}{ 35 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&2&-3&-50&75\\& & 10& \color{blue}{35} & \\ \hline &2&\color{blue}{7}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -50 } + \color{orangered}{ 35 } = \color{orangered}{ -15 } $
$$ \begin{array}{c|rrrr}5&2&-3&\color{orangered}{ -50 }&75\\& & 10& \color{orangered}{35} & \\ \hline &2&7&\color{orangered}{-15}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -15 \right) } = \color{blue}{ -75 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&2&-3&-50&75\\& & 10& 35& \color{blue}{-75} \\ \hline &2&7&\color{blue}{-15}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 75 } + \color{orangered}{ \left( -75 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}5&2&-3&-50&\color{orangered}{ 75 }\\& & 10& 35& \color{orangered}{-75} \\ \hline &\color{blue}{2}&\color{blue}{7}&\color{blue}{-15}&\color{orangered}{0} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 0 }\right) $.