Divide using synthetic division $$ ( 3x^{3}-15x^{2}+7x-25 ) \div ( x-5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}5&3&-15&7&-25\\& & 15& 0& \color{black}{35} \\ \hline &\color{blue}{3}&\color{blue}{0}&\color{blue}{7}&\color{orangered}{10} \end{array} $$The solution is:
$$ \frac{ 3x^{3}-15x^{2}+7x-25 }{ x-5 } = \color{blue}{3x^{2}+7} ~+~ \frac{ \color{red}{ 10 } }{ x-5 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{5}&3&-15&7&-25\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}5&\color{orangered}{ 3 }&-15&7&-25\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 3 } = \color{blue}{ 15 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&3&-15&7&-25\\& & \color{blue}{15} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -15 } + \color{orangered}{ 15 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}5&3&\color{orangered}{ -15 }&7&-25\\& & \color{orangered}{15} & & \\ \hline &3&\color{orangered}{0}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&3&-15&7&-25\\& & 15& \color{blue}{0} & \\ \hline &3&\color{blue}{0}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 7 } + \color{orangered}{ 0 } = \color{orangered}{ 7 } $
$$ \begin{array}{c|rrrr}5&3&-15&\color{orangered}{ 7 }&-25\\& & 15& \color{orangered}{0} & \\ \hline &3&0&\color{orangered}{7}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 7 } = \color{blue}{ 35 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&3&-15&7&-25\\& & 15& 0& \color{blue}{35} \\ \hline &3&0&\color{blue}{7}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -25 } + \color{orangered}{ 35 } = \color{orangered}{ 10 } $
$$ \begin{array}{c|rrrr}5&3&-15&7&\color{orangered}{ -25 }\\& & 15& 0& \color{orangered}{35} \\ \hline &\color{blue}{3}&\color{blue}{0}&\color{blue}{7}&\color{orangered}{10} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{2}+7 } $ with a remainder of $ \color{red}{ 10 } $.