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}{ 1 } = \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}{ 1 } = \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}{ \left( -12 \right) } = \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}{ \left( -12 \right) } = \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}{ \left( -64 \right) } = \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}{ \left( -64 \right) } = \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 } $.