Divide using synthetic division $$ ( 3x^{2}-8x+10 ) \div ( x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-5&3&-8&10\\& & -15& \color{black}{115} \\ \hline &\color{blue}{3}&\color{blue}{-23}&\color{orangered}{125} \end{array} $$The solution is:
$$ \frac{ 3x^{2}-8x+10 }{ x+5 } = \color{blue}{3x-23} ~+~ \frac{ \color{red}{ 125 } }{ x+5 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{-5}&3&-8&10\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-5&\color{orangered}{ 3 }&-8&10\\& & & \\ \hline &\color{orangered}{3}&& \end{array} $$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 } $.
$$ \begin{array}{c|rrr}\color{blue}{-5}&3&-8&10\\& & \color{blue}{-15} & \\ \hline &\color{blue}{3}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ \left( -15 \right) } = \color{orangered}{ -23 } $
$$ \begin{array}{c|rrr}-5&3&\color{orangered}{ -8 }&10\\& & \color{orangered}{-15} & \\ \hline &3&\color{orangered}{-23}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -23 \right) } = \color{blue}{ 115 } $.
$$ \begin{array}{c|rrr}\color{blue}{-5}&3&-8&10\\& & -15& \color{blue}{115} \\ \hline &3&\color{blue}{-23}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 10 } + \color{orangered}{ 115 } = \color{orangered}{ 125 } $
$$ \begin{array}{c|rrr}-5&3&-8&\color{orangered}{ 10 }\\& & -15& \color{orangered}{115} \\ \hline &\color{blue}{3}&\color{blue}{-23}&\color{orangered}{125} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x-23 } $ with a remainder of $ \color{red}{ 125 } $.