Divide using synthetic division $$ ( 8x^{3}+11 ) \div ( -10x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-5&8&0&0&11\\& & -40& 200& \color{black}{-1000} \\ \hline &\color{blue}{8}&\color{blue}{-40}&\color{blue}{200}&\color{orangered}{-989} \end{array} $$The solution is:
$$ \frac{ 8x^{3}+11 }{ x+5 } = \color{blue}{8x^{2}-40x+200} \color{red}{~-~} \frac{ \color{red}{ 989 } }{ x+5 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&8&0&0&11\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-5&\color{orangered}{ 8 }&0&0&11\\& & & & \\ \hline &\color{orangered}{8}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 8 } = \color{blue}{ -40 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&8&0&0&11\\& & \color{blue}{-40} & & \\ \hline &\color{blue}{8}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -40 \right) } = \color{orangered}{ -40 } $
$$ \begin{array}{c|rrrr}-5&8&\color{orangered}{ 0 }&0&11\\& & \color{orangered}{-40} & & \\ \hline &8&\color{orangered}{-40}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -40 \right) } = \color{blue}{ 200 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&8&0&0&11\\& & -40& \color{blue}{200} & \\ \hline &8&\color{blue}{-40}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 200 } = \color{orangered}{ 200 } $
$$ \begin{array}{c|rrrr}-5&8&0&\color{orangered}{ 0 }&11\\& & -40& \color{orangered}{200} & \\ \hline &8&-40&\color{orangered}{200}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 200 } = \color{blue}{ -1000 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&8&0&0&11\\& & -40& 200& \color{blue}{-1000} \\ \hline &8&-40&\color{blue}{200}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 11 } + \color{orangered}{ \left( -1000 \right) } = \color{orangered}{ -989 } $
$$ \begin{array}{c|rrrr}-5&8&0&0&\color{orangered}{ 11 }\\& & -40& 200& \color{orangered}{-1000} \\ \hline &\color{blue}{8}&\color{blue}{-40}&\color{blue}{200}&\color{orangered}{-989} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 8x^{2}-40x+200 } $ with a remainder of $ \color{red}{ -989 } $.