Divide using synthetic division $$ ( x^{6}-7x^{4}-12x^{2}-26x+24 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}3&1&0&-7&0&-12&-26&24\\& & 3& 9& 6& 18& 18& \color{black}{-24} \\ \hline &\color{blue}{1}&\color{blue}{3}&\color{blue}{2}&\color{blue}{6}&\color{blue}{6}&\color{blue}{-8}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{6}-7x^{4}-12x^{2}-26x+24 }{ x-3 } = \color{blue}{x^{5}+3x^{4}+2x^{3}+6x^{2}+6x-8} $$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.
$$ \begin{array}{c|rrrrrrr}\color{blue}{3}&1&0&-7&0&-12&-26&24\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}3&\color{orangered}{ 1 }&0&-7&0&-12&-26&24\\& & & & & & & \\ \hline &\color{orangered}{1}&&&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{3}&1&0&-7&0&-12&-26&24\\& & \color{blue}{3} & & & & & \\ \hline &\color{blue}{1}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 3 } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrrrrr}3&1&\color{orangered}{ 0 }&-7&0&-12&-26&24\\& & \color{orangered}{3} & & & & & \\ \hline &1&\color{orangered}{3}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 3 } = \color{blue}{ 9 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{3}&1&0&-7&0&-12&-26&24\\& & 3& \color{blue}{9} & & & & \\ \hline &1&\color{blue}{3}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ 9 } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrrrrr}3&1&0&\color{orangered}{ -7 }&0&-12&-26&24\\& & 3& \color{orangered}{9} & & & & \\ \hline &1&3&\color{orangered}{2}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{3}&1&0&-7&0&-12&-26&24\\& & 3& 9& \color{blue}{6} & & & \\ \hline &1&3&\color{blue}{2}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 6 } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrrrrr}3&1&0&-7&\color{orangered}{ 0 }&-12&-26&24\\& & 3& 9& \color{orangered}{6} & & & \\ \hline &1&3&2&\color{orangered}{6}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 6 } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{3}&1&0&-7&0&-12&-26&24\\& & 3& 9& 6& \color{blue}{18} & & \\ \hline &1&3&2&\color{blue}{6}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -12 } + \color{orangered}{ 18 } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrrrrr}3&1&0&-7&0&\color{orangered}{ -12 }&-26&24\\& & 3& 9& 6& \color{orangered}{18} & & \\ \hline &1&3&2&6&\color{orangered}{6}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 6 } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{3}&1&0&-7&0&-12&-26&24\\& & 3& 9& 6& 18& \color{blue}{18} & \\ \hline &1&3&2&6&\color{blue}{6}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -26 } + \color{orangered}{ 18 } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrrrrrr}3&1&0&-7&0&-12&\color{orangered}{ -26 }&24\\& & 3& 9& 6& 18& \color{orangered}{18} & \\ \hline &1&3&2&6&6&\color{orangered}{-8}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ -24 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{3}&1&0&-7&0&-12&-26&24\\& & 3& 9& 6& 18& 18& \color{blue}{-24} \\ \hline &1&3&2&6&6&\color{blue}{-8}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 24 } + \color{orangered}{ \left( -24 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrrr}3&1&0&-7&0&-12&-26&\color{orangered}{ 24 }\\& & 3& 9& 6& 18& 18& \color{orangered}{-24} \\ \hline &\color{blue}{1}&\color{blue}{3}&\color{blue}{2}&\color{blue}{6}&\color{blue}{6}&\color{blue}{-8}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{5}+3x^{4}+2x^{3}+6x^{2}+6x-8 } $ with a remainder of $ \color{red}{ 0 } $.