Find remainder when $ 8x^{2}-17x-17 $ is divided by $ x-3 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}3&8&-17&-17\\& & 24& \color{black}{21} \\ \hline &\color{blue}{8}&\color{blue}{7}&\color{orangered}{4} \end{array} $$The remainder when $ 8x^{2}-17x-17 $ is divided by $ x-3 $ is $ \, \color{red}{ 4 } $.
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.
$$ \begin{array}{c|rrr}\color{blue}{3}&8&-17&-17\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}3&\color{orangered}{ 8 }&-17&-17\\& & & \\ \hline &\color{orangered}{8}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 8 } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrr}\color{blue}{3}&8&-17&-17\\& & \color{blue}{24} & \\ \hline &\color{blue}{8}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ 24 } = \color{orangered}{ 7 } $
$$ \begin{array}{c|rrr}3&8&\color{orangered}{ -17 }&-17\\& & \color{orangered}{24} & \\ \hline &8&\color{orangered}{7}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 7 } = \color{blue}{ 21 } $.
$$ \begin{array}{c|rrr}\color{blue}{3}&8&-17&-17\\& & 24& \color{blue}{21} \\ \hline &8&\color{blue}{7}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ 21 } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrr}3&8&-17&\color{orangered}{ -17 }\\& & 24& \color{orangered}{21} \\ \hline &\color{blue}{8}&\color{blue}{7}&\color{orangered}{4} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 4 }\right) $.