Divide using synthetic division $$ ( 104x^{3}-14x^{2}+17x-10 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}2&104&-14&17&-10\\& & 208& 388& \color{black}{810} \\ \hline &\color{blue}{104}&\color{blue}{194}&\color{blue}{405}&\color{orangered}{800} \end{array} $$The solution is:
$$ \frac{ 104x^{3}-14x^{2}+17x-10 }{ x-2 } = \color{blue}{104x^{2}+194x+405} ~+~ \frac{ \color{red}{ 800 } }{ x-2 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{2}&104&-14&17&-10\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}2&\color{orangered}{ 104 }&-14&17&-10\\& & & & \\ \hline &\color{orangered}{104}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 104 } = \color{blue}{ 208 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&104&-14&17&-10\\& & \color{blue}{208} & & \\ \hline &\color{blue}{104}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -14 } + \color{orangered}{ 208 } = \color{orangered}{ 194 } $
$$ \begin{array}{c|rrrr}2&104&\color{orangered}{ -14 }&17&-10\\& & \color{orangered}{208} & & \\ \hline &104&\color{orangered}{194}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 194 } = \color{blue}{ 388 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&104&-14&17&-10\\& & 208& \color{blue}{388} & \\ \hline &104&\color{blue}{194}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 17 } + \color{orangered}{ 388 } = \color{orangered}{ 405 } $
$$ \begin{array}{c|rrrr}2&104&-14&\color{orangered}{ 17 }&-10\\& & 208& \color{orangered}{388} & \\ \hline &104&194&\color{orangered}{405}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 405 } = \color{blue}{ 810 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&104&-14&17&-10\\& & 208& 388& \color{blue}{810} \\ \hline &104&194&\color{blue}{405}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ 810 } = \color{orangered}{ 800 } $
$$ \begin{array}{c|rrrr}2&104&-14&17&\color{orangered}{ -10 }\\& & 208& 388& \color{orangered}{810} \\ \hline &\color{blue}{104}&\color{blue}{194}&\color{blue}{405}&\color{orangered}{800} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 104x^{2}+194x+405 } $ with a remainder of $ \color{red}{ 800 } $.