Find remainder when $ 6x^{3}-21x^{2}+25x-8 $ is divided by $ x-1 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}1&6&-21&25&-8\\& & 6& -15& \color{black}{10} \\ \hline &\color{blue}{6}&\color{blue}{-15}&\color{blue}{10}&\color{orangered}{2} \end{array} $$The remainder when $ 6x^{3}-21x^{2}+25x-8 $ is divided by $ x-1 $ is $ \, \color{red}{ 2 } $.
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.
$$ \begin{array}{c|rrrr}\color{blue}{1}&6&-21&25&-8\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}1&\color{orangered}{ 6 }&-21&25&-8\\& & & & \\ \hline &\color{orangered}{6}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 6 } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrr}\color{blue}{1}&6&-21&25&-8\\& & \color{blue}{6} & & \\ \hline &\color{blue}{6}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -21 } + \color{orangered}{ 6 } = \color{orangered}{ -15 } $
$$ \begin{array}{c|rrrr}1&6&\color{orangered}{ -21 }&25&-8\\& & \color{orangered}{6} & & \\ \hline &6&\color{orangered}{-15}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \left( -15 \right) } = \color{blue}{ -15 } $.
$$ \begin{array}{c|rrrr}\color{blue}{1}&6&-21&25&-8\\& & 6& \color{blue}{-15} & \\ \hline &6&\color{blue}{-15}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 25 } + \color{orangered}{ \left( -15 \right) } = \color{orangered}{ 10 } $
$$ \begin{array}{c|rrrr}1&6&-21&\color{orangered}{ 25 }&-8\\& & 6& \color{orangered}{-15} & \\ \hline &6&-15&\color{orangered}{10}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 10 } = \color{blue}{ 10 } $.
$$ \begin{array}{c|rrrr}\color{blue}{1}&6&-21&25&-8\\& & 6& -15& \color{blue}{10} \\ \hline &6&-15&\color{blue}{10}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ 10 } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrr}1&6&-21&25&\color{orangered}{ -8 }\\& & 6& -15& \color{orangered}{10} \\ \hline &\color{blue}{6}&\color{blue}{-15}&\color{blue}{10}&\color{orangered}{2} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 2 }\right) $.