Find remainder when $ 2x^{4}+7x^{2}-7x-15 $ is divided by $ 2x-3 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}3&2&0&7&-7&-15\\& & 6& 18& 75& \color{black}{204} \\ \hline &\color{blue}{2}&\color{blue}{6}&\color{blue}{25}&\color{blue}{68}&\color{orangered}{189} \end{array} $$The remainder when $ 2x^{4}+7x^{2}-7x-15 $ is divided by $ x-3 $ is $ \, \color{red}{ 189 } $.
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 -3 = 0 $ ( $ x = \color{blue}{ 3 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&2&0&7&-7&-15\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}3&\color{orangered}{ 2 }&0&7&-7&-15\\& & & & & \\ \hline &\color{orangered}{2}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 2 } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&2&0&7&-7&-15\\& & \color{blue}{6} & & & \\ \hline &\color{blue}{2}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 6 } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrrr}3&2&\color{orangered}{ 0 }&7&-7&-15\\& & \color{orangered}{6} & & & \\ \hline &2&\color{orangered}{6}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 6 } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&2&0&7&-7&-15\\& & 6& \color{blue}{18} & & \\ \hline &2&\color{blue}{6}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 7 } + \color{orangered}{ 18 } = \color{orangered}{ 25 } $
$$ \begin{array}{c|rrrrr}3&2&0&\color{orangered}{ 7 }&-7&-15\\& & 6& \color{orangered}{18} & & \\ \hline &2&6&\color{orangered}{25}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 25 } = \color{blue}{ 75 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&2&0&7&-7&-15\\& & 6& 18& \color{blue}{75} & \\ \hline &2&6&\color{blue}{25}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ 75 } = \color{orangered}{ 68 } $
$$ \begin{array}{c|rrrrr}3&2&0&7&\color{orangered}{ -7 }&-15\\& & 6& 18& \color{orangered}{75} & \\ \hline &2&6&25&\color{orangered}{68}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 68 } = \color{blue}{ 204 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&2&0&7&-7&-15\\& & 6& 18& 75& \color{blue}{204} \\ \hline &2&6&25&\color{blue}{68}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -15 } + \color{orangered}{ 204 } = \color{orangered}{ 189 } $
$$ \begin{array}{c|rrrrr}3&2&0&7&-7&\color{orangered}{ -15 }\\& & 6& 18& 75& \color{orangered}{204} \\ \hline &\color{blue}{2}&\color{blue}{6}&\color{blue}{25}&\color{blue}{68}&\color{orangered}{189} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 189 }\right) $.