Divide using synthetic division $$ ( x^{3}+4x^{2}-21x-62 ) \div ( x+6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-6&1&4&-21&-62\\& & -6& 12& \color{black}{54} \\ \hline &\color{blue}{1}&\color{blue}{-2}&\color{blue}{-9}&\color{orangered}{-8} \end{array} $$The solution is:
$$ \frac{ x^{3}+4x^{2}-21x-62 }{ x+6 } = \color{blue}{x^{2}-2x-9} \color{red}{~-~} \frac{ \color{red}{ 8 } }{ x+6 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 6 = 0 $ ( $ x = \color{blue}{ -6 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&4&-21&-62\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-6&\color{orangered}{ 1 }&4&-21&-62\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 1 } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&4&-21&-62\\& & \color{blue}{-6} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrr}-6&1&\color{orangered}{ 4 }&-21&-62\\& & \color{orangered}{-6} & & \\ \hline &1&\color{orangered}{-2}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ 12 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&4&-21&-62\\& & -6& \color{blue}{12} & \\ \hline &1&\color{blue}{-2}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -21 } + \color{orangered}{ 12 } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrr}-6&1&4&\color{orangered}{ -21 }&-62\\& & -6& \color{orangered}{12} & \\ \hline &1&-2&\color{orangered}{-9}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ 54 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&4&-21&-62\\& & -6& 12& \color{blue}{54} \\ \hline &1&-2&\color{blue}{-9}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -62 } + \color{orangered}{ 54 } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrrr}-6&1&4&-21&\color{orangered}{ -62 }\\& & -6& 12& \color{orangered}{54} \\ \hline &\color{blue}{1}&\color{blue}{-2}&\color{blue}{-9}&\color{orangered}{-8} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}-2x-9 } $ with a remainder of $ \color{red}{ -8 } $.