Divide using synthetic division $$ ( 2x^{3}+9x^{2}-2x-24 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-2&2&9&-2&-24\\& & -4& -10& \color{black}{24} \\ \hline &\color{blue}{2}&\color{blue}{5}&\color{blue}{-12}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 2x^{3}+9x^{2}-2x-24 }{ x+2 } = \color{blue}{2x^{2}+5x-12} $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&2&9&-2&-24\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-2&\color{orangered}{ 2 }&9&-2&-24\\& & & & \\ \hline &\color{orangered}{2}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 2 } = \color{blue}{ -4 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&2&9&-2&-24\\& & \color{blue}{-4} & & \\ \hline &\color{blue}{2}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrrr}-2&2&\color{orangered}{ 9 }&-2&-24\\& & \color{orangered}{-4} & & \\ \hline &2&\color{orangered}{5}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 5 } = \color{blue}{ -10 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&2&9&-2&-24\\& & -4& \color{blue}{-10} & \\ \hline &2&\color{blue}{5}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ \left( -10 \right) } = \color{orangered}{ -12 } $
$$ \begin{array}{c|rrrr}-2&2&9&\color{orangered}{ -2 }&-24\\& & -4& \color{orangered}{-10} & \\ \hline &2&5&\color{orangered}{-12}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -12 \right) } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&2&9&-2&-24\\& & -4& -10& \color{blue}{24} \\ \hline &2&5&\color{blue}{-12}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -24 } + \color{orangered}{ 24 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}-2&2&9&-2&\color{orangered}{ -24 }\\& & -4& -10& \color{orangered}{24} \\ \hline &\color{blue}{2}&\color{blue}{5}&\color{blue}{-12}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{2}+5x-12 } $ with a remainder of $ \color{red}{ 0 } $.