Divide using synthetic division $$ ( 25x^{3}-41x+8 ) \div ( 5x-1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}1&25&0&-41&8\\& & 25& 25& \color{black}{-16} \\ \hline &\color{blue}{25}&\color{blue}{25}&\color{blue}{-16}&\color{orangered}{-8} \end{array} $$The solution is:
$$ \frac{ 25x^{3}-41x+8 }{ x-1 } = \color{blue}{25x^{2}+25x-16} \color{red}{~-~} \frac{ \color{red}{ 8 } }{ x-1 } $$Explanation
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.
$$ \begin{array}{c|rrrr}\color{blue}{1}&25&0&-41&8\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}1&\color{orangered}{ 25 }&0&-41&8\\& & & & \\ \hline &\color{orangered}{25}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 25 } = \color{blue}{ 25 } $.
$$ \begin{array}{c|rrrr}\color{blue}{1}&25&0&-41&8\\& & \color{blue}{25} & & \\ \hline &\color{blue}{25}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 25 } = \color{orangered}{ 25 } $
$$ \begin{array}{c|rrrr}1&25&\color{orangered}{ 0 }&-41&8\\& & \color{orangered}{25} & & \\ \hline &25&\color{orangered}{25}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 25 } = \color{blue}{ 25 } $.
$$ \begin{array}{c|rrrr}\color{blue}{1}&25&0&-41&8\\& & 25& \color{blue}{25} & \\ \hline &25&\color{blue}{25}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -41 } + \color{orangered}{ 25 } = \color{orangered}{ -16 } $
$$ \begin{array}{c|rrrr}1&25&0&\color{orangered}{ -41 }&8\\& & 25& \color{orangered}{25} & \\ \hline &25&25&\color{orangered}{-16}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \left( -16 \right) } = \color{blue}{ -16 } $.
$$ \begin{array}{c|rrrr}\color{blue}{1}&25&0&-41&8\\& & 25& 25& \color{blue}{-16} \\ \hline &25&25&\color{blue}{-16}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ \left( -16 \right) } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrrr}1&25&0&-41&\color{orangered}{ 8 }\\& & 25& 25& \color{orangered}{-16} \\ \hline &\color{blue}{25}&\color{blue}{25}&\color{blue}{-16}&\color{orangered}{-8} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 25x^{2}+25x-16 } $ with a remainder of $ \color{red}{ -8 } $.