Divide using synthetic division $$ ( x^{4}-7x^{2}-18 ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-3&1&0&-7&0&-18\\& & -3& 9& -6& \color{black}{18} \\ \hline &\color{blue}{1}&\color{blue}{-3}&\color{blue}{2}&\color{blue}{-6}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{4}-7x^{2}-18 }{ x+3 } = \color{blue}{x^{3}-3x^{2}+2x-6} $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&0&-7&0&-18\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-3&\color{orangered}{ 1 }&0&-7&0&-18\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&0&-7&0&-18\\& & \color{blue}{-3} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -3 \right) } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrrrr}-3&1&\color{orangered}{ 0 }&-7&0&-18\\& & \color{orangered}{-3} & & & \\ \hline &1&\color{orangered}{-3}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&0&-7&0&-18\\& & -3& \color{blue}{9} & & \\ \hline &1&\color{blue}{-3}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ 9 } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrrr}-3&1&0&\color{orangered}{ -7 }&0&-18\\& & -3& \color{orangered}{9} & & \\ \hline &1&-3&\color{orangered}{2}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 2 } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&0&-7&0&-18\\& & -3& 9& \color{blue}{-6} & \\ \hline &1&-3&\color{blue}{2}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrrr}-3&1&0&-7&\color{orangered}{ 0 }&-18\\& & -3& 9& \color{orangered}{-6} & \\ \hline &1&-3&2&\color{orangered}{-6}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&0&-7&0&-18\\& & -3& 9& -6& \color{blue}{18} \\ \hline &1&-3&2&\color{blue}{-6}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -18 } + \color{orangered}{ 18 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}-3&1&0&-7&0&\color{orangered}{ -18 }\\& & -3& 9& -6& \color{orangered}{18} \\ \hline &\color{blue}{1}&\color{blue}{-3}&\color{blue}{2}&\color{blue}{-6}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}-3x^{2}+2x-6 } $ with a remainder of $ \color{red}{ 0 } $.