Divide using synthetic division $$ ( 8x^{3}+x^{2}-4x-2 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-2&8&1&-4&-2\\& & -16& 30& \color{black}{-52} \\ \hline &\color{blue}{8}&\color{blue}{-15}&\color{blue}{26}&\color{orangered}{-54} \end{array} $$The solution is:
$$ \frac{ 8x^{3}+x^{2}-4x-2 }{ x+2 } = \color{blue}{8x^{2}-15x+26} \color{red}{~-~} \frac{ \color{red}{ 54 } }{ x+2 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 2 = 0 $ ( $ x = \color{blue}{ -2 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&8&1&-4&-2\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-2&\color{orangered}{ 8 }&1&-4&-2\\& & & & \\ \hline &\color{orangered}{8}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 8 } = \color{blue}{ -16 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&8&1&-4&-2\\& & \color{blue}{-16} & & \\ \hline &\color{blue}{8}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ \left( -16 \right) } = \color{orangered}{ -15 } $
$$ \begin{array}{c|rrrr}-2&8&\color{orangered}{ 1 }&-4&-2\\& & \color{orangered}{-16} & & \\ \hline &8&\color{orangered}{-15}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -15 \right) } = \color{blue}{ 30 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&8&1&-4&-2\\& & -16& \color{blue}{30} & \\ \hline &8&\color{blue}{-15}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 30 } = \color{orangered}{ 26 } $
$$ \begin{array}{c|rrrr}-2&8&1&\color{orangered}{ -4 }&-2\\& & -16& \color{orangered}{30} & \\ \hline &8&-15&\color{orangered}{26}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 26 } = \color{blue}{ -52 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&8&1&-4&-2\\& & -16& 30& \color{blue}{-52} \\ \hline &8&-15&\color{blue}{26}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ \left( -52 \right) } = \color{orangered}{ -54 } $
$$ \begin{array}{c|rrrr}-2&8&1&-4&\color{orangered}{ -2 }\\& & -16& 30& \color{orangered}{-52} \\ \hline &\color{blue}{8}&\color{blue}{-15}&\color{blue}{26}&\color{orangered}{-54} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 8x^{2}-15x+26 } $ with a remainder of $ \color{red}{ -54 } $.