Divide using synthetic division $$ ( 3x^{3}-5x^{2}-26x-8 ) \div ( 3x+1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-1&3&-5&-26&-8\\& & -3& 8& \color{black}{18} \\ \hline &\color{blue}{3}&\color{blue}{-8}&\color{blue}{-18}&\color{orangered}{10} \end{array} $$The solution is:
$$ \frac{ 3x^{3}-5x^{2}-26x-8 }{ x+1 } = \color{blue}{3x^{2}-8x-18} ~+~ \frac{ \color{red}{ 10 } }{ x+1 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 1 = 0 $ ( $ x = \color{blue}{ -1 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&3&-5&-26&-8\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-1&\color{orangered}{ 3 }&-5&-26&-8\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 3 } = \color{blue}{ -3 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&3&-5&-26&-8\\& & \color{blue}{-3} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ \left( -3 \right) } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrrr}-1&3&\color{orangered}{ -5 }&-26&-8\\& & \color{orangered}{-3} & & \\ \hline &3&\color{orangered}{-8}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&3&-5&-26&-8\\& & -3& \color{blue}{8} & \\ \hline &3&\color{blue}{-8}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -26 } + \color{orangered}{ 8 } = \color{orangered}{ -18 } $
$$ \begin{array}{c|rrrr}-1&3&-5&\color{orangered}{ -26 }&-8\\& & -3& \color{orangered}{8} & \\ \hline &3&-8&\color{orangered}{-18}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -18 \right) } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&3&-5&-26&-8\\& & -3& 8& \color{blue}{18} \\ \hline &3&-8&\color{blue}{-18}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ 18 } = \color{orangered}{ 10 } $
$$ \begin{array}{c|rrrr}-1&3&-5&-26&\color{orangered}{ -8 }\\& & -3& 8& \color{orangered}{18} \\ \hline &\color{blue}{3}&\color{blue}{-8}&\color{blue}{-18}&\color{orangered}{10} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{2}-8x-18 } $ with a remainder of $ \color{red}{ 10 } $.