Divide using synthetic division $$ ( x^{6}-13x^{4}-52x^{2}+64 ) \div ( x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}-5&1&0&-13&0&-52&0&64\\& & -5& 25& -60& 300& -1240& \color{black}{6200} \\ \hline &\color{blue}{1}&\color{blue}{-5}&\color{blue}{12}&\color{blue}{-60}&\color{blue}{248}&\color{blue}{-1240}&\color{orangered}{6264} \end{array} $$The solution is:
$$ \frac{ x^{6}-13x^{4}-52x^{2}+64 }{ x+5 } = \color{blue}{x^{5}-5x^{4}+12x^{3}-60x^{2}+248x-1240} ~+~ \frac{ \color{red}{ 6264 } }{ x+5 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 5 = 0 $ ( $ x = \color{blue}{ -5 } $ ) at the left.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-5}&1&0&-13&0&-52&0&64\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}-5&\color{orangered}{ 1 }&0&-13&0&-52&0&64\\& & & & & & & \\ \hline &\color{orangered}{1}&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 1 } = \color{blue}{ -5 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-5}&1&0&-13&0&-52&0&64\\& & \color{blue}{-5} & & & & & \\ \hline &\color{blue}{1}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -5 \right) } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrrrrrr}-5&1&\color{orangered}{ 0 }&-13&0&-52&0&64\\& & \color{orangered}{-5} & & & & & \\ \hline &1&\color{orangered}{-5}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ 25 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-5}&1&0&-13&0&-52&0&64\\& & -5& \color{blue}{25} & & & & \\ \hline &1&\color{blue}{-5}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -13 } + \color{orangered}{ 25 } = \color{orangered}{ 12 } $
$$ \begin{array}{c|rrrrrrr}-5&1&0&\color{orangered}{ -13 }&0&-52&0&64\\& & -5& \color{orangered}{25} & & & & \\ \hline &1&-5&\color{orangered}{12}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 12 } = \color{blue}{ -60 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-5}&1&0&-13&0&-52&0&64\\& & -5& 25& \color{blue}{-60} & & & \\ \hline &1&-5&\color{blue}{12}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -60 \right) } = \color{orangered}{ -60 } $
$$ \begin{array}{c|rrrrrrr}-5&1&0&-13&\color{orangered}{ 0 }&-52&0&64\\& & -5& 25& \color{orangered}{-60} & & & \\ \hline &1&-5&12&\color{orangered}{-60}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -60 \right) } = \color{blue}{ 300 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-5}&1&0&-13&0&-52&0&64\\& & -5& 25& -60& \color{blue}{300} & & \\ \hline &1&-5&12&\color{blue}{-60}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -52 } + \color{orangered}{ 300 } = \color{orangered}{ 248 } $
$$ \begin{array}{c|rrrrrrr}-5&1&0&-13&0&\color{orangered}{ -52 }&0&64\\& & -5& 25& -60& \color{orangered}{300} & & \\ \hline &1&-5&12&-60&\color{orangered}{248}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 248 } = \color{blue}{ -1240 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-5}&1&0&-13&0&-52&0&64\\& & -5& 25& -60& 300& \color{blue}{-1240} & \\ \hline &1&-5&12&-60&\color{blue}{248}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -1240 \right) } = \color{orangered}{ -1240 } $
$$ \begin{array}{c|rrrrrrr}-5&1&0&-13&0&-52&\color{orangered}{ 0 }&64\\& & -5& 25& -60& 300& \color{orangered}{-1240} & \\ \hline &1&-5&12&-60&248&\color{orangered}{-1240}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -1240 \right) } = \color{blue}{ 6200 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-5}&1&0&-13&0&-52&0&64\\& & -5& 25& -60& 300& -1240& \color{blue}{6200} \\ \hline &1&-5&12&-60&248&\color{blue}{-1240}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 64 } + \color{orangered}{ 6200 } = \color{orangered}{ 6264 } $
$$ \begin{array}{c|rrrrrrr}-5&1&0&-13&0&-52&0&\color{orangered}{ 64 }\\& & -5& 25& -60& 300& -1240& \color{orangered}{6200} \\ \hline &\color{blue}{1}&\color{blue}{-5}&\color{blue}{12}&\color{blue}{-60}&\color{blue}{248}&\color{blue}{-1240}&\color{orangered}{6264} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{5}-5x^{4}+12x^{3}-60x^{2}+248x-1240 } $ with a remainder of $ \color{red}{ 6264 } $.