Find remainder when $ 3x^{3}+x^{2}-22x-25 $ is divided by $ x-2 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}2&3&1&-22&-25\\& & 6& 14& \color{black}{-16} \\ \hline &\color{blue}{3}&\color{blue}{7}&\color{blue}{-8}&\color{orangered}{-41} \end{array} $$The remainder when $ 3x^{3}+x^{2}-22x-25 $ is divided by $ x-2 $ is $ \, \color{red}{ -41 } $.
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 -2 = 0 $ ( $ x = \color{blue}{ 2 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{2}&3&1&-22&-25\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}2&\color{orangered}{ 3 }&1&-22&-25\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 3 } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&3&1&-22&-25\\& & \color{blue}{6} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 6 } = \color{orangered}{ 7 } $
$$ \begin{array}{c|rrrr}2&3&\color{orangered}{ 1 }&-22&-25\\& & \color{orangered}{6} & & \\ \hline &3&\color{orangered}{7}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 7 } = \color{blue}{ 14 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&3&1&-22&-25\\& & 6& \color{blue}{14} & \\ \hline &3&\color{blue}{7}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -22 } + \color{orangered}{ 14 } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrrr}2&3&1&\color{orangered}{ -22 }&-25\\& & 6& \color{orangered}{14} & \\ \hline &3&7&\color{orangered}{-8}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ -16 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&3&1&-22&-25\\& & 6& 14& \color{blue}{-16} \\ \hline &3&7&\color{blue}{-8}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -25 } + \color{orangered}{ \left( -16 \right) } = \color{orangered}{ -41 } $
$$ \begin{array}{c|rrrr}2&3&1&-22&\color{orangered}{ -25 }\\& & 6& 14& \color{orangered}{-16} \\ \hline &\color{blue}{3}&\color{blue}{7}&\color{blue}{-8}&\color{orangered}{-41} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -41 }\right) $.