Divide using synthetic division $$ ( x^{6}-13x^{4}-52x^{2}+64 ) \div ( x+1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}-1&1&0&-13&0&-52&0&64\\& & -1& 1& 12& -12& 64& \color{black}{-64} \\ \hline &\color{blue}{1}&\color{blue}{-1}&\color{blue}{-12}&\color{blue}{12}&\color{blue}{-64}&\color{blue}{64}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{6}-13x^{4}-52x^{2}+64 }{ x+1 } = \color{blue}{x^{5}-x^{4}-12x^{3}+12x^{2}-64x+64} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 1 = 0 $ ( $ x = \color{blue}{ -1 } $ ) at the left.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-1}&1&0&-13&0&-52&0&64\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}-1&\color{orangered}{ 1 }&0&-13&0&-52&0&64\\& & & & & & & \\ \hline &\color{orangered}{1}&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 1 } = \color{blue}{ -1 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-1}&1&0&-13&0&-52&0&64\\& & \color{blue}{-1} & & & & & \\ \hline &\color{blue}{1}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -1 \right) } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrrrrr}-1&1&\color{orangered}{ 0 }&-13&0&-52&0&64\\& & \color{orangered}{-1} & & & & & \\ \hline &1&\color{orangered}{-1}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 1 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-1}&1&0&-13&0&-52&0&64\\& & -1& \color{blue}{1} & & & & \\ \hline &1&\color{blue}{-1}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -13 } + \color{orangered}{ 1 } = \color{orangered}{ -12 } $
$$ \begin{array}{c|rrrrrrr}-1&1&0&\color{orangered}{ -13 }&0&-52&0&64\\& & -1& \color{orangered}{1} & & & & \\ \hline &1&-1&\color{orangered}{-12}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -12 \right) } = \color{blue}{ 12 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-1}&1&0&-13&0&-52&0&64\\& & -1& 1& \color{blue}{12} & & & \\ \hline &1&-1&\color{blue}{-12}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 12 } = \color{orangered}{ 12 } $
$$ \begin{array}{c|rrrrrrr}-1&1&0&-13&\color{orangered}{ 0 }&-52&0&64\\& & -1& 1& \color{orangered}{12} & & & \\ \hline &1&-1&-12&\color{orangered}{12}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 12 } = \color{blue}{ -12 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-1}&1&0&-13&0&-52&0&64\\& & -1& 1& 12& \color{blue}{-12} & & \\ \hline &1&-1&-12&\color{blue}{12}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -52 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ -64 } $
$$ \begin{array}{c|rrrrrrr}-1&1&0&-13&0&\color{orangered}{ -52 }&0&64\\& & -1& 1& 12& \color{orangered}{-12} & & \\ \hline &1&-1&-12&12&\color{orangered}{-64}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -64 \right) } = \color{blue}{ 64 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-1}&1&0&-13&0&-52&0&64\\& & -1& 1& 12& -12& \color{blue}{64} & \\ \hline &1&-1&-12&12&\color{blue}{-64}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 64 } = \color{orangered}{ 64 } $
$$ \begin{array}{c|rrrrrrr}-1&1&0&-13&0&-52&\color{orangered}{ 0 }&64\\& & -1& 1& 12& -12& \color{orangered}{64} & \\ \hline &1&-1&-12&12&-64&\color{orangered}{64}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 64 } = \color{blue}{ -64 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-1}&1&0&-13&0&-52&0&64\\& & -1& 1& 12& -12& 64& \color{blue}{-64} \\ \hline &1&-1&-12&12&-64&\color{blue}{64}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 64 } + \color{orangered}{ \left( -64 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrrr}-1&1&0&-13&0&-52&0&\color{orangered}{ 64 }\\& & -1& 1& 12& -12& 64& \color{orangered}{-64} \\ \hline &\color{blue}{1}&\color{blue}{-1}&\color{blue}{-12}&\color{blue}{12}&\color{blue}{-64}&\color{blue}{64}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{5}-x^{4}-12x^{3}+12x^{2}-64x+64 } $ with a remainder of $ \color{red}{ 0 } $.