Divide using synthetic division $$ ( 6x^{3}-2x^{2}-15x+5 ) \div ( 2x-5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}5&6&-2&-15&5\\& & 30& 140& \color{black}{625} \\ \hline &\color{blue}{6}&\color{blue}{28}&\color{blue}{125}&\color{orangered}{630} \end{array} $$The solution is:
$$ \frac{ 6x^{3}-2x^{2}-15x+5 }{ x-5 } = \color{blue}{6x^{2}+28x+125} ~+~ \frac{ \color{red}{ 630 } }{ x-5 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{5}&6&-2&-15&5\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}5&\color{orangered}{ 6 }&-2&-15&5\\& & & & \\ \hline &\color{orangered}{6}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 6 } = \color{blue}{ 30 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&6&-2&-15&5\\& & \color{blue}{30} & & \\ \hline &\color{blue}{6}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 30 } = \color{orangered}{ 28 } $
$$ \begin{array}{c|rrrr}5&6&\color{orangered}{ -2 }&-15&5\\& & \color{orangered}{30} & & \\ \hline &6&\color{orangered}{28}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 28 } = \color{blue}{ 140 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&6&-2&-15&5\\& & 30& \color{blue}{140} & \\ \hline &6&\color{blue}{28}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -15 } + \color{orangered}{ 140 } = \color{orangered}{ 125 } $
$$ \begin{array}{c|rrrr}5&6&-2&\color{orangered}{ -15 }&5\\& & 30& \color{orangered}{140} & \\ \hline &6&28&\color{orangered}{125}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 125 } = \color{blue}{ 625 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&6&-2&-15&5\\& & 30& 140& \color{blue}{625} \\ \hline &6&28&\color{blue}{125}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 625 } = \color{orangered}{ 630 } $
$$ \begin{array}{c|rrrr}5&6&-2&-15&\color{orangered}{ 5 }\\& & 30& 140& \color{orangered}{625} \\ \hline &\color{blue}{6}&\color{blue}{28}&\color{blue}{125}&\color{orangered}{630} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 6x^{2}+28x+125 } $ with a remainder of $ \color{red}{ 630 } $.