Find remainder when $ 2x^{4}-5x^{3}+x^{2}-4 $ is divided by $ x+2 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-2&2&-5&1&0&-4\\& & -4& 18& -38& \color{black}{76} \\ \hline &\color{blue}{2}&\color{blue}{-9}&\color{blue}{19}&\color{blue}{-38}&\color{orangered}{72} \end{array} $$The remainder when $ 2x^{4}-5x^{3}+x^{2}-4 $ is divided by $ x+2 $ is $ \, \color{red}{ 72 } $.
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.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&2&-5&1&0&-4\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-2&\color{orangered}{ 2 }&-5&1&0&-4\\& & & & & \\ \hline &\color{orangered}{2}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&2&-5&1&0&-4\\& & \color{blue}{-4} & & & \\ \hline &\color{blue}{2}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrrr}-2&2&\color{orangered}{ -5 }&1&0&-4\\& & \color{orangered}{-4} & & & \\ \hline &2&\color{orangered}{-9}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&2&-5&1&0&-4\\& & -4& \color{blue}{18} & & \\ \hline &2&\color{blue}{-9}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 18 } = \color{orangered}{ 19 } $
$$ \begin{array}{c|rrrrr}-2&2&-5&\color{orangered}{ 1 }&0&-4\\& & -4& \color{orangered}{18} & & \\ \hline &2&-9&\color{orangered}{19}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 19 } = \color{blue}{ -38 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&2&-5&1&0&-4\\& & -4& 18& \color{blue}{-38} & \\ \hline &2&-9&\color{blue}{19}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -38 \right) } = \color{orangered}{ -38 } $
$$ \begin{array}{c|rrrrr}-2&2&-5&1&\color{orangered}{ 0 }&-4\\& & -4& 18& \color{orangered}{-38} & \\ \hline &2&-9&19&\color{orangered}{-38}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -38 \right) } = \color{blue}{ 76 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&2&-5&1&0&-4\\& & -4& 18& -38& \color{blue}{76} \\ \hline &2&-9&19&\color{blue}{-38}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 76 } = \color{orangered}{ 72 } $
$$ \begin{array}{c|rrrrr}-2&2&-5&1&0&\color{orangered}{ -4 }\\& & -4& 18& -38& \color{orangered}{76} \\ \hline &\color{blue}{2}&\color{blue}{-9}&\color{blue}{19}&\color{blue}{-38}&\color{orangered}{72} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 72 }\right) $.