Divide using synthetic division $$ ( x^{5}+39x-36 ) \div ( x-4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}4&1&0&0&0&39&-36\\& & 4& 16& 64& 256& \color{black}{1180} \\ \hline &\color{blue}{1}&\color{blue}{4}&\color{blue}{16}&\color{blue}{64}&\color{blue}{295}&\color{orangered}{1144} \end{array} $$The solution is:
$$ \frac{ x^{5}+39x-36 }{ x-4 } = \color{blue}{x^{4}+4x^{3}+16x^{2}+64x+295} ~+~ \frac{ \color{red}{ 1144 } }{ x-4 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -4 = 0 $ ( $ x = \color{blue}{ 4 } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{4}&1&0&0&0&39&-36\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}4&\color{orangered}{ 1 }&0&0&0&39&-36\\& & & & & & \\ \hline &\color{orangered}{1}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 1 } = \color{blue}{ 4 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{4}&1&0&0&0&39&-36\\& & \color{blue}{4} & & & & \\ \hline &\color{blue}{1}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 4 } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrrrr}4&1&\color{orangered}{ 0 }&0&0&39&-36\\& & \color{orangered}{4} & & & & \\ \hline &1&\color{orangered}{4}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 4 } = \color{blue}{ 16 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{4}&1&0&0&0&39&-36\\& & 4& \color{blue}{16} & & & \\ \hline &1&\color{blue}{4}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 16 } = \color{orangered}{ 16 } $
$$ \begin{array}{c|rrrrrr}4&1&0&\color{orangered}{ 0 }&0&39&-36\\& & 4& \color{orangered}{16} & & & \\ \hline &1&4&\color{orangered}{16}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 16 } = \color{blue}{ 64 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{4}&1&0&0&0&39&-36\\& & 4& 16& \color{blue}{64} & & \\ \hline &1&4&\color{blue}{16}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 64 } = \color{orangered}{ 64 } $
$$ \begin{array}{c|rrrrrr}4&1&0&0&\color{orangered}{ 0 }&39&-36\\& & 4& 16& \color{orangered}{64} & & \\ \hline &1&4&16&\color{orangered}{64}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 64 } = \color{blue}{ 256 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{4}&1&0&0&0&39&-36\\& & 4& 16& 64& \color{blue}{256} & \\ \hline &1&4&16&\color{blue}{64}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 39 } + \color{orangered}{ 256 } = \color{orangered}{ 295 } $
$$ \begin{array}{c|rrrrrr}4&1&0&0&0&\color{orangered}{ 39 }&-36\\& & 4& 16& 64& \color{orangered}{256} & \\ \hline &1&4&16&64&\color{orangered}{295}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 295 } = \color{blue}{ 1180 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{4}&1&0&0&0&39&-36\\& & 4& 16& 64& 256& \color{blue}{1180} \\ \hline &1&4&16&64&\color{blue}{295}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -36 } + \color{orangered}{ 1180 } = \color{orangered}{ 1144 } $
$$ \begin{array}{c|rrrrrr}4&1&0&0&0&39&\color{orangered}{ -36 }\\& & 4& 16& 64& 256& \color{orangered}{1180} \\ \hline &\color{blue}{1}&\color{blue}{4}&\color{blue}{16}&\color{blue}{64}&\color{blue}{295}&\color{orangered}{1144} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{4}+4x^{3}+16x^{2}+64x+295 } $ with a remainder of $ \color{red}{ 1144 } $.