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