Divide using synthetic division $$ ( 2x^{4}+x^{3}+39x^{2}+136x-78 ) \div ( x-1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}1&2&1&39&136&-78\\& & 2& 3& 42& \color{black}{178} \\ \hline &\color{blue}{2}&\color{blue}{3}&\color{blue}{42}&\color{blue}{178}&\color{orangered}{100} \end{array} $$The solution is:
$$ \frac{ 2x^{4}+x^{3}+39x^{2}+136x-78 }{ x-1 } = \color{blue}{2x^{3}+3x^{2}+42x+178} ~+~ \frac{ \color{red}{ 100 } }{ x-1 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{1}&2&1&39&136&-78\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}1&\color{orangered}{ 2 }&1&39&136&-78\\& & & & & \\ \hline &\color{orangered}{2}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 2 } = \color{blue}{ 2 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{1}&2&1&39&136&-78\\& & \color{blue}{2} & & & \\ \hline &\color{blue}{2}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 2 } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrrr}1&2&\color{orangered}{ 1 }&39&136&-78\\& & \color{orangered}{2} & & & \\ \hline &2&\color{orangered}{3}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 3 } = \color{blue}{ 3 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{1}&2&1&39&136&-78\\& & 2& \color{blue}{3} & & \\ \hline &2&\color{blue}{3}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 39 } + \color{orangered}{ 3 } = \color{orangered}{ 42 } $
$$ \begin{array}{c|rrrrr}1&2&1&\color{orangered}{ 39 }&136&-78\\& & 2& \color{orangered}{3} & & \\ \hline &2&3&\color{orangered}{42}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 42 } = \color{blue}{ 42 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{1}&2&1&39&136&-78\\& & 2& 3& \color{blue}{42} & \\ \hline &2&3&\color{blue}{42}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 136 } + \color{orangered}{ 42 } = \color{orangered}{ 178 } $
$$ \begin{array}{c|rrrrr}1&2&1&39&\color{orangered}{ 136 }&-78\\& & 2& 3& \color{orangered}{42} & \\ \hline &2&3&42&\color{orangered}{178}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 178 } = \color{blue}{ 178 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{1}&2&1&39&136&-78\\& & 2& 3& 42& \color{blue}{178} \\ \hline &2&3&42&\color{blue}{178}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -78 } + \color{orangered}{ 178 } = \color{orangered}{ 100 } $
$$ \begin{array}{c|rrrrr}1&2&1&39&136&\color{orangered}{ -78 }\\& & 2& 3& 42& \color{orangered}{178} \\ \hline &\color{blue}{2}&\color{blue}{3}&\color{blue}{42}&\color{blue}{178}&\color{orangered}{100} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{3}+3x^{2}+42x+178 } $ with a remainder of $ \color{red}{ 100 } $.