Divide using synthetic division $$ ( x^{3}+11x^{2}+31x+21 ) \div ( x+7 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-7&1&11&31&21\\& & -7& -28& \color{black}{-21} \\ \hline &\color{blue}{1}&\color{blue}{4}&\color{blue}{3}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{3}+11x^{2}+31x+21 }{ x+7 } = \color{blue}{x^{2}+4x+3} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 7 = 0 $ ( $ x = \color{blue}{ -7 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-7}&1&11&31&21\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-7&\color{orangered}{ 1 }&11&31&21\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ 1 } = \color{blue}{ -7 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-7}&1&11&31&21\\& & \color{blue}{-7} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 11 } + \color{orangered}{ \left( -7 \right) } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrr}-7&1&\color{orangered}{ 11 }&31&21\\& & \color{orangered}{-7} & & \\ \hline &1&\color{orangered}{4}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ 4 } = \color{blue}{ -28 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-7}&1&11&31&21\\& & -7& \color{blue}{-28} & \\ \hline &1&\color{blue}{4}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 31 } + \color{orangered}{ \left( -28 \right) } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrr}-7&1&11&\color{orangered}{ 31 }&21\\& & -7& \color{orangered}{-28} & \\ \hline &1&4&\color{orangered}{3}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ 3 } = \color{blue}{ -21 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-7}&1&11&31&21\\& & -7& -28& \color{blue}{-21} \\ \hline &1&4&\color{blue}{3}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 21 } + \color{orangered}{ \left( -21 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}-7&1&11&31&\color{orangered}{ 21 }\\& & -7& -28& \color{orangered}{-21} \\ \hline &\color{blue}{1}&\color{blue}{4}&\color{blue}{3}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}+4x+3 } $ with a remainder of $ \color{red}{ 0 } $.