Divide using synthetic division $$ ( 5x^{3}-x^{2}-18x+8 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-2&5&-1&-18&8\\& & -10& 22& \color{black}{-8} \\ \hline &\color{blue}{5}&\color{blue}{-11}&\color{blue}{4}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 5x^{3}-x^{2}-18x+8 }{ x+2 } = \color{blue}{5x^{2}-11x+4} $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&5&-1&-18&8\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-2&\color{orangered}{ 5 }&-1&-18&8\\& & & & \\ \hline &\color{orangered}{5}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 5 } = \color{blue}{ -10 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&5&-1&-18&8\\& & \color{blue}{-10} & & \\ \hline &\color{blue}{5}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ \left( -10 \right) } = \color{orangered}{ -11 } $
$$ \begin{array}{c|rrrr}-2&5&\color{orangered}{ -1 }&-18&8\\& & \color{orangered}{-10} & & \\ \hline &5&\color{orangered}{-11}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -11 \right) } = \color{blue}{ 22 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&5&-1&-18&8\\& & -10& \color{blue}{22} & \\ \hline &5&\color{blue}{-11}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -18 } + \color{orangered}{ 22 } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrr}-2&5&-1&\color{orangered}{ -18 }&8\\& & -10& \color{orangered}{22} & \\ \hline &5&-11&\color{orangered}{4}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 4 } = \color{blue}{ -8 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&5&-1&-18&8\\& & -10& 22& \color{blue}{-8} \\ \hline &5&-11&\color{blue}{4}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}-2&5&-1&-18&\color{orangered}{ 8 }\\& & -10& 22& \color{orangered}{-8} \\ \hline &\color{blue}{5}&\color{blue}{-11}&\color{blue}{4}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 5x^{2}-11x+4 } $ with a remainder of $ \color{red}{ 0 } $.