Find remainder when $ 4x^{3}-20x^{2}+25x-125 $ is divided by $ x-6 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}6&4&-20&25&-125\\& & 24& 24& \color{black}{294} \\ \hline &\color{blue}{4}&\color{blue}{4}&\color{blue}{49}&\color{orangered}{169} \end{array} $$The remainder when $ 4x^{3}-20x^{2}+25x-125 $ is divided by $ x-6 $ is $ \, \color{red}{ 169 } $.
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 -6 = 0 $ ( $ x = \color{blue}{ 6 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{6}&4&-20&25&-125\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}6&\color{orangered}{ 4 }&-20&25&-125\\& & & & \\ \hline &\color{orangered}{4}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 4 } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrr}\color{blue}{6}&4&-20&25&-125\\& & \color{blue}{24} & & \\ \hline &\color{blue}{4}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -20 } + \color{orangered}{ 24 } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrr}6&4&\color{orangered}{ -20 }&25&-125\\& & \color{orangered}{24} & & \\ \hline &4&\color{orangered}{4}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 4 } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrr}\color{blue}{6}&4&-20&25&-125\\& & 24& \color{blue}{24} & \\ \hline &4&\color{blue}{4}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 25 } + \color{orangered}{ 24 } = \color{orangered}{ 49 } $
$$ \begin{array}{c|rrrr}6&4&-20&\color{orangered}{ 25 }&-125\\& & 24& \color{orangered}{24} & \\ \hline &4&4&\color{orangered}{49}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 49 } = \color{blue}{ 294 } $.
$$ \begin{array}{c|rrrr}\color{blue}{6}&4&-20&25&-125\\& & 24& 24& \color{blue}{294} \\ \hline &4&4&\color{blue}{49}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -125 } + \color{orangered}{ 294 } = \color{orangered}{ 169 } $
$$ \begin{array}{c|rrrr}6&4&-20&25&\color{orangered}{ -125 }\\& & 24& 24& \color{orangered}{294} \\ \hline &\color{blue}{4}&\color{blue}{4}&\color{blue}{49}&\color{orangered}{169} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 169 }\right) $.