Divide using synthetic division $$ ( 2x^{7}-19x^{2}+18x+8 ) \div ( 2x-5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrrr}5&2&0&0&0&0&-19&18&8\\& & 10& 50& 250& 1250& 6250& 31155& \color{black}{155865} \\ \hline &\color{blue}{2}&\color{blue}{10}&\color{blue}{50}&\color{blue}{250}&\color{blue}{1250}&\color{blue}{6231}&\color{blue}{31173}&\color{orangered}{155873} \end{array} $$The solution is:
$$ \frac{ 2x^{7}-19x^{2}+18x+8 }{ x-5 } = \color{blue}{2x^{6}+10x^{5}+50x^{4}+250x^{3}+1250x^{2}+6231x+31173} ~+~ \frac{ \color{red}{ 155873 } }{ x-5 } $$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.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{5}&2&0&0&0&0&-19&18&8\\& & & & & & & & \\ \hline &&&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrrr}5&\color{orangered}{ 2 }&0&0&0&0&-19&18&8\\& & & & & & & & \\ \hline &\color{orangered}{2}&&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 2 } = \color{blue}{ 10 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{5}&2&0&0&0&0&-19&18&8\\& & \color{blue}{10} & & & & & & \\ \hline &\color{blue}{2}&&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 10 } = \color{orangered}{ 10 } $
$$ \begin{array}{c|rrrrrrrr}5&2&\color{orangered}{ 0 }&0&0&0&-19&18&8\\& & \color{orangered}{10} & & & & & & \\ \hline &2&\color{orangered}{10}&&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 10 } = \color{blue}{ 50 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{5}&2&0&0&0&0&-19&18&8\\& & 10& \color{blue}{50} & & & & & \\ \hline &2&\color{blue}{10}&&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 50 } = \color{orangered}{ 50 } $
$$ \begin{array}{c|rrrrrrrr}5&2&0&\color{orangered}{ 0 }&0&0&-19&18&8\\& & 10& \color{orangered}{50} & & & & & \\ \hline &2&10&\color{orangered}{50}&&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 50 } = \color{blue}{ 250 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{5}&2&0&0&0&0&-19&18&8\\& & 10& 50& \color{blue}{250} & & & & \\ \hline &2&10&\color{blue}{50}&&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 250 } = \color{orangered}{ 250 } $
$$ \begin{array}{c|rrrrrrrr}5&2&0&0&\color{orangered}{ 0 }&0&-19&18&8\\& & 10& 50& \color{orangered}{250} & & & & \\ \hline &2&10&50&\color{orangered}{250}&&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 250 } = \color{blue}{ 1250 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{5}&2&0&0&0&0&-19&18&8\\& & 10& 50& 250& \color{blue}{1250} & & & \\ \hline &2&10&50&\color{blue}{250}&&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 1250 } = \color{orangered}{ 1250 } $
$$ \begin{array}{c|rrrrrrrr}5&2&0&0&0&\color{orangered}{ 0 }&-19&18&8\\& & 10& 50& 250& \color{orangered}{1250} & & & \\ \hline &2&10&50&250&\color{orangered}{1250}&&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 1250 } = \color{blue}{ 6250 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{5}&2&0&0&0&0&-19&18&8\\& & 10& 50& 250& 1250& \color{blue}{6250} & & \\ \hline &2&10&50&250&\color{blue}{1250}&&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -19 } + \color{orangered}{ 6250 } = \color{orangered}{ 6231 } $
$$ \begin{array}{c|rrrrrrrr}5&2&0&0&0&0&\color{orangered}{ -19 }&18&8\\& & 10& 50& 250& 1250& \color{orangered}{6250} & & \\ \hline &2&10&50&250&1250&\color{orangered}{6231}&& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 6231 } = \color{blue}{ 31155 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{5}&2&0&0&0&0&-19&18&8\\& & 10& 50& 250& 1250& 6250& \color{blue}{31155} & \\ \hline &2&10&50&250&1250&\color{blue}{6231}&& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 18 } + \color{orangered}{ 31155 } = \color{orangered}{ 31173 } $
$$ \begin{array}{c|rrrrrrrr}5&2&0&0&0&0&-19&\color{orangered}{ 18 }&8\\& & 10& 50& 250& 1250& 6250& \color{orangered}{31155} & \\ \hline &2&10&50&250&1250&6231&\color{orangered}{31173}& \end{array} $$Step 14 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 31173 } = \color{blue}{ 155865 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{5}&2&0&0&0&0&-19&18&8\\& & 10& 50& 250& 1250& 6250& 31155& \color{blue}{155865} \\ \hline &2&10&50&250&1250&6231&\color{blue}{31173}& \end{array} $$Step 15 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ 155865 } = \color{orangered}{ 155873 } $
$$ \begin{array}{c|rrrrrrrr}5&2&0&0&0&0&-19&18&\color{orangered}{ 8 }\\& & 10& 50& 250& 1250& 6250& 31155& \color{orangered}{155865} \\ \hline &\color{blue}{2}&\color{blue}{10}&\color{blue}{50}&\color{blue}{250}&\color{blue}{1250}&\color{blue}{6231}&\color{blue}{31173}&\color{orangered}{155873} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{6}+10x^{5}+50x^{4}+250x^{3}+1250x^{2}+6231x+31173 } $ with a remainder of $ \color{red}{ 155873 } $.