Divide using synthetic division $$ ( x^{3}-15x^{2}+75x-125 ) \div ( x-5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}5&1&-15&75&-125\\& & 5& -50& \color{black}{125} \\ \hline &\color{blue}{1}&\color{blue}{-10}&\color{blue}{25}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{3}-15x^{2}+75x-125 }{ x-5 } = \color{blue}{x^{2}-10x+25} $$Explanation
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.
$$ \begin{array}{c|rrrr}\color{blue}{5}&1&-15&75&-125\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}5&\color{orangered}{ 1 }&-15&75&-125\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 1 } = \color{blue}{ 5 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&1&-15&75&-125\\& & \color{blue}{5} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -15 } + \color{orangered}{ 5 } = \color{orangered}{ -10 } $
$$ \begin{array}{c|rrrr}5&1&\color{orangered}{ -15 }&75&-125\\& & \color{orangered}{5} & & \\ \hline &1&\color{orangered}{-10}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -10 \right) } = \color{blue}{ -50 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&1&-15&75&-125\\& & 5& \color{blue}{-50} & \\ \hline &1&\color{blue}{-10}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 75 } + \color{orangered}{ \left( -50 \right) } = \color{orangered}{ 25 } $
$$ \begin{array}{c|rrrr}5&1&-15&\color{orangered}{ 75 }&-125\\& & 5& \color{orangered}{-50} & \\ \hline &1&-10&\color{orangered}{25}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 25 } = \color{blue}{ 125 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&1&-15&75&-125\\& & 5& -50& \color{blue}{125} \\ \hline &1&-10&\color{blue}{25}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -125 } + \color{orangered}{ 125 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}5&1&-15&75&\color{orangered}{ -125 }\\& & 5& -50& \color{orangered}{125} \\ \hline &\color{blue}{1}&\color{blue}{-10}&\color{blue}{25}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}-10x+25 } $ with a remainder of $ \color{red}{ 0 } $.