Find remainder when $ 4x^{2}-22x-25 $ is divided by $ x-2 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}2&4&-22&-25\\& & 8& \color{black}{-28} \\ \hline &\color{blue}{4}&\color{blue}{-14}&\color{orangered}{-53} \end{array} $$The remainder when $ 4x^{2}-22x-25 $ is divided by $ x-2 $ is $ \, \color{red}{ -53 } $.
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.
$$ \begin{array}{c|rrr}\color{blue}{2}&4&-22&-25\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}2&\color{orangered}{ 4 }&-22&-25\\& & & \\ \hline &\color{orangered}{4}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 4 } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrr}\color{blue}{2}&4&-22&-25\\& & \color{blue}{8} & \\ \hline &\color{blue}{4}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -22 } + \color{orangered}{ 8 } = \color{orangered}{ -14 } $
$$ \begin{array}{c|rrr}2&4&\color{orangered}{ -22 }&-25\\& & \color{orangered}{8} & \\ \hline &4&\color{orangered}{-14}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -14 \right) } = \color{blue}{ -28 } $.
$$ \begin{array}{c|rrr}\color{blue}{2}&4&-22&-25\\& & 8& \color{blue}{-28} \\ \hline &4&\color{blue}{-14}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -25 } + \color{orangered}{ \left( -28 \right) } = \color{orangered}{ -53 } $
$$ \begin{array}{c|rrr}2&4&-22&\color{orangered}{ -25 }\\& & 8& \color{orangered}{-28} \\ \hline &\color{blue}{4}&\color{blue}{-14}&\color{orangered}{-53} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -53 }\right) $.