Divide using synthetic division $$ ( 2x^{2}+5 ) \div ( x-29 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}29&2&0&5\\& & 58& \color{black}{1682} \\ \hline &\color{blue}{2}&\color{blue}{58}&\color{orangered}{1687} \end{array} $$The solution is:
$$ \frac{ 2x^{2}+5 }{ x-29 } = \color{blue}{2x+58} ~+~ \frac{ \color{red}{ 1687 } }{ x-29 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -29 = 0 $ ( $ x = \color{blue}{ 29 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{29}&2&0&5\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}29&\color{orangered}{ 2 }&0&5\\& & & \\ \hline &\color{orangered}{2}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 29 } \cdot \color{blue}{ 2 } = \color{blue}{ 58 } $.
$$ \begin{array}{c|rrr}\color{blue}{29}&2&0&5\\& & \color{blue}{58} & \\ \hline &\color{blue}{2}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 58 } = \color{orangered}{ 58 } $
$$ \begin{array}{c|rrr}29&2&\color{orangered}{ 0 }&5\\& & \color{orangered}{58} & \\ \hline &2&\color{orangered}{58}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 29 } \cdot \color{blue}{ 58 } = \color{blue}{ 1682 } $.
$$ \begin{array}{c|rrr}\color{blue}{29}&2&0&5\\& & 58& \color{blue}{1682} \\ \hline &2&\color{blue}{58}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 1682 } = \color{orangered}{ 1687 } $
$$ \begin{array}{c|rrr}29&2&0&\color{orangered}{ 5 }\\& & 58& \color{orangered}{1682} \\ \hline &\color{blue}{2}&\color{blue}{58}&\color{orangered}{1687} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x+58 } $ with a remainder of $ \color{red}{ 1687 } $.