Divide using synthetic division $$ ( 4x^{3}-17x^{2}+14x+8 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}2&4&-17&14&8\\& & 8& -18& \color{black}{-8} \\ \hline &\color{blue}{4}&\color{blue}{-9}&\color{blue}{-4}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 4x^{3}-17x^{2}+14x+8 }{ x-2 } = \color{blue}{4x^{2}-9x-4} $$Explanation
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.
$$ \begin{array}{c|rrrr}\color{blue}{2}&4&-17&14&8\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}2&\color{orangered}{ 4 }&-17&14&8\\& & & & \\ \hline &\color{orangered}{4}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&4&-17&14&8\\& & \color{blue}{8} & & \\ \hline &\color{blue}{4}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ 8 } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrr}2&4&\color{orangered}{ -17 }&14&8\\& & \color{orangered}{8} & & \\ \hline &4&\color{orangered}{-9}&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&4&-17&14&8\\& & 8& \color{blue}{-18} & \\ \hline &4&\color{blue}{-9}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 14 } + \color{orangered}{ \left( -18 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrr}2&4&-17&\color{orangered}{ 14 }&8\\& & 8& \color{orangered}{-18} & \\ \hline &4&-9&\color{orangered}{-4}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ -8 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&4&-17&14&8\\& & 8& -18& \color{blue}{-8} \\ \hline &4&-9&\color{blue}{-4}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}2&4&-17&14&\color{orangered}{ 8 }\\& & 8& -18& \color{orangered}{-8} \\ \hline &\color{blue}{4}&\color{blue}{-9}&\color{blue}{-4}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x^{2}-9x-4 } $ with a remainder of $ \color{red}{ 0 } $.