Divide using synthetic division $$ ( x^{4}-16x^{2}-14x+24 ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-3&1&0&-16&-14&24\\& & -3& 9& 21& \color{black}{-21} \\ \hline &\color{blue}{1}&\color{blue}{-3}&\color{blue}{-7}&\color{blue}{7}&\color{orangered}{3} \end{array} $$The solution is:
$$ \frac{ x^{4}-16x^{2}-14x+24 }{ x+3 } = \color{blue}{x^{3}-3x^{2}-7x+7} ~+~ \frac{ \color{red}{ 3 } }{ x+3 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 3 = 0 $ ( $ x = \color{blue}{ -3 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&0&-16&-14&24\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-3&\color{orangered}{ 1 }&0&-16&-14&24\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 1 } = \color{blue}{ -3 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&0&-16&-14&24\\& & \color{blue}{-3} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -3 \right) } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrrrr}-3&1&\color{orangered}{ 0 }&-16&-14&24\\& & \color{orangered}{-3} & & & \\ \hline &1&\color{orangered}{-3}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ 9 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&0&-16&-14&24\\& & -3& \color{blue}{9} & & \\ \hline &1&\color{blue}{-3}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -16 } + \color{orangered}{ 9 } = \color{orangered}{ -7 } $
$$ \begin{array}{c|rrrrr}-3&1&0&\color{orangered}{ -16 }&-14&24\\& & -3& \color{orangered}{9} & & \\ \hline &1&-3&\color{orangered}{-7}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -7 \right) } = \color{blue}{ 21 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&0&-16&-14&24\\& & -3& 9& \color{blue}{21} & \\ \hline &1&-3&\color{blue}{-7}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -14 } + \color{orangered}{ 21 } = \color{orangered}{ 7 } $
$$ \begin{array}{c|rrrrr}-3&1&0&-16&\color{orangered}{ -14 }&24\\& & -3& 9& \color{orangered}{21} & \\ \hline &1&-3&-7&\color{orangered}{7}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 7 } = \color{blue}{ -21 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&0&-16&-14&24\\& & -3& 9& 21& \color{blue}{-21} \\ \hline &1&-3&-7&\color{blue}{7}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 24 } + \color{orangered}{ \left( -21 \right) } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrrr}-3&1&0&-16&-14&\color{orangered}{ 24 }\\& & -3& 9& 21& \color{orangered}{-21} \\ \hline &\color{blue}{1}&\color{blue}{-3}&\color{blue}{-7}&\color{blue}{7}&\color{orangered}{3} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}-3x^{2}-7x+7 } $ with a remainder of $ \color{red}{ 3 } $.