Divide using synthetic division $$ ( 6x^{3}+25x^{2}-29x-20 ) \div ( 2x+1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-1&6&25&-29&-20\\& & -6& -19& \color{black}{48} \\ \hline &\color{blue}{6}&\color{blue}{19}&\color{blue}{-48}&\color{orangered}{28} \end{array} $$The solution is:
$$ \frac{ 6x^{3}+25x^{2}-29x-20 }{ x+1 } = \color{blue}{6x^{2}+19x-48} ~+~ \frac{ \color{red}{ 28 } }{ x+1 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&6&25&-29&-20\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-1&\color{orangered}{ 6 }&25&-29&-20\\& & & & \\ \hline &\color{orangered}{6}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 6 } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&6&25&-29&-20\\& & \color{blue}{-6} & & \\ \hline &\color{blue}{6}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 25 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ 19 } $
$$ \begin{array}{c|rrrr}-1&6&\color{orangered}{ 25 }&-29&-20\\& & \color{orangered}{-6} & & \\ \hline &6&\color{orangered}{19}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 19 } = \color{blue}{ -19 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&6&25&-29&-20\\& & -6& \color{blue}{-19} & \\ \hline &6&\color{blue}{19}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -29 } + \color{orangered}{ \left( -19 \right) } = \color{orangered}{ -48 } $
$$ \begin{array}{c|rrrr}-1&6&25&\color{orangered}{ -29 }&-20\\& & -6& \color{orangered}{-19} & \\ \hline &6&19&\color{orangered}{-48}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -48 \right) } = \color{blue}{ 48 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&6&25&-29&-20\\& & -6& -19& \color{blue}{48} \\ \hline &6&19&\color{blue}{-48}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -20 } + \color{orangered}{ 48 } = \color{orangered}{ 28 } $
$$ \begin{array}{c|rrrr}-1&6&25&-29&\color{orangered}{ -20 }\\& & -6& -19& \color{orangered}{48} \\ \hline &\color{blue}{6}&\color{blue}{19}&\color{blue}{-48}&\color{orangered}{28} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 6x^{2}+19x-48 } $ with a remainder of $ \color{red}{ 28 } $.