Divide using synthetic division $$ ( 3x^{3}-7x^{2}-21x-10 ) \div ( 3x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}2&3&-7&-21&-10\\& & 6& -2& \color{black}{-46} \\ \hline &\color{blue}{3}&\color{blue}{-1}&\color{blue}{-23}&\color{orangered}{-56} \end{array} $$The solution is:
$$ \frac{ 3x^{3}-7x^{2}-21x-10 }{ x-2 } = \color{blue}{3x^{2}-x-23} \color{red}{~-~} \frac{ \color{red}{ 56 } }{ x-2 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{2}&3&-7&-21&-10\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}2&\color{orangered}{ 3 }&-7&-21&-10\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&3&-7&-21&-10\\& & \color{blue}{6} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ 6 } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrr}2&3&\color{orangered}{ -7 }&-21&-10\\& & \color{orangered}{6} & & \\ \hline &3&\color{orangered}{-1}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ -2 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&3&-7&-21&-10\\& & 6& \color{blue}{-2} & \\ \hline &3&\color{blue}{-1}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -21 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ -23 } $
$$ \begin{array}{c|rrrr}2&3&-7&\color{orangered}{ -21 }&-10\\& & 6& \color{orangered}{-2} & \\ \hline &3&-1&\color{orangered}{-23}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -23 \right) } = \color{blue}{ -46 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&3&-7&-21&-10\\& & 6& -2& \color{blue}{-46} \\ \hline &3&-1&\color{blue}{-23}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ \left( -46 \right) } = \color{orangered}{ -56 } $
$$ \begin{array}{c|rrrr}2&3&-7&-21&\color{orangered}{ -10 }\\& & 6& -2& \color{orangered}{-46} \\ \hline &\color{blue}{3}&\color{blue}{-1}&\color{blue}{-23}&\color{orangered}{-56} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{2}-x-23 } $ with a remainder of $ \color{red}{ -56 } $.