Find remainder when $ 2x^{2}-9x+30 $ is divided by $ x+2 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-2&2&-9&30\\& & -4& \color{black}{26} \\ \hline &\color{blue}{2}&\color{blue}{-13}&\color{orangered}{56} \end{array} $$The remainder when $ 2x^{2}-9x+30 $ is divided by $ x+2 $ is $ \, \color{red}{ 56 } $.
Explanation
We can find remainder using synthetic division method.
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 2 = 0 $ ( $ x = \color{blue}{ -2 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{-2}&2&-9&30\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-2&\color{orangered}{ 2 }&-9&30\\& & & \\ \hline &\color{orangered}{2}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 2 } = \color{blue}{ -4 } $.
$$ \begin{array}{c|rrr}\color{blue}{-2}&2&-9&30\\& & \color{blue}{-4} & \\ \hline &\color{blue}{2}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ -13 } $
$$ \begin{array}{c|rrr}-2&2&\color{orangered}{ -9 }&30\\& & \color{orangered}{-4} & \\ \hline &2&\color{orangered}{-13}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -13 \right) } = \color{blue}{ 26 } $.
$$ \begin{array}{c|rrr}\color{blue}{-2}&2&-9&30\\& & -4& \color{blue}{26} \\ \hline &2&\color{blue}{-13}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 30 } + \color{orangered}{ 26 } = \color{orangered}{ 56 } $
$$ \begin{array}{c|rrr}-2&2&-9&\color{orangered}{ 30 }\\& & -4& \color{orangered}{26} \\ \hline &\color{blue}{2}&\color{blue}{-13}&\color{orangered}{56} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 56 }\right) $.