Divide using synthetic division $$ ( 3x^{2}-7x-10 ) \div ( x-5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}5&3&-7&-10\\& & 15& \color{black}{40} \\ \hline &\color{blue}{3}&\color{blue}{8}&\color{orangered}{30} \end{array} $$The solution is:
$$ \frac{ 3x^{2}-7x-10 }{ x-5 } = \color{blue}{3x+8} ~+~ \frac{ \color{red}{ 30 } }{ x-5 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{5}&3&-7&-10\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}5&\color{orangered}{ 3 }&-7&-10\\& & & \\ \hline &\color{orangered}{3}&& \end{array} $$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 } $.
$$ \begin{array}{c|rrr}\color{blue}{5}&3&-7&-10\\& & \color{blue}{15} & \\ \hline &\color{blue}{3}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ 15 } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrr}5&3&\color{orangered}{ -7 }&-10\\& & \color{orangered}{15} & \\ \hline &3&\color{orangered}{8}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 8 } = \color{blue}{ 40 } $.
$$ \begin{array}{c|rrr}\color{blue}{5}&3&-7&-10\\& & 15& \color{blue}{40} \\ \hline &3&\color{blue}{8}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ 40 } = \color{orangered}{ 30 } $
$$ \begin{array}{c|rrr}5&3&-7&\color{orangered}{ -10 }\\& & 15& \color{orangered}{40} \\ \hline &\color{blue}{3}&\color{blue}{8}&\color{orangered}{30} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x+8 } $ with a remainder of $ \color{red}{ 30 } $.