Divide using synthetic division $$ ( 3x^{3}+14x^{2}-40x-17 ) \div ( 3x-7 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}7&3&14&-40&-17\\& & 21& 245& \color{black}{1435} \\ \hline &\color{blue}{3}&\color{blue}{35}&\color{blue}{205}&\color{orangered}{1418} \end{array} $$The solution is:
$$ \frac{ 3x^{3}+14x^{2}-40x-17 }{ x-7 } = \color{blue}{3x^{2}+35x+205} ~+~ \frac{ \color{red}{ 1418 } }{ x-7 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -7 = 0 $ ( $ x = \color{blue}{ 7 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{7}&3&14&-40&-17\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}7&\color{orangered}{ 3 }&14&-40&-17\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ 3 } = \color{blue}{ 21 } $.
$$ \begin{array}{c|rrrr}\color{blue}{7}&3&14&-40&-17\\& & \color{blue}{21} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 14 } + \color{orangered}{ 21 } = \color{orangered}{ 35 } $
$$ \begin{array}{c|rrrr}7&3&\color{orangered}{ 14 }&-40&-17\\& & \color{orangered}{21} & & \\ \hline &3&\color{orangered}{35}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ 35 } = \color{blue}{ 245 } $.
$$ \begin{array}{c|rrrr}\color{blue}{7}&3&14&-40&-17\\& & 21& \color{blue}{245} & \\ \hline &3&\color{blue}{35}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -40 } + \color{orangered}{ 245 } = \color{orangered}{ 205 } $
$$ \begin{array}{c|rrrr}7&3&14&\color{orangered}{ -40 }&-17\\& & 21& \color{orangered}{245} & \\ \hline &3&35&\color{orangered}{205}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ 205 } = \color{blue}{ 1435 } $.
$$ \begin{array}{c|rrrr}\color{blue}{7}&3&14&-40&-17\\& & 21& 245& \color{blue}{1435} \\ \hline &3&35&\color{blue}{205}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ 1435 } = \color{orangered}{ 1418 } $
$$ \begin{array}{c|rrrr}7&3&14&-40&\color{orangered}{ -17 }\\& & 21& 245& \color{orangered}{1435} \\ \hline &\color{blue}{3}&\color{blue}{35}&\color{blue}{205}&\color{orangered}{1418} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{2}+35x+205 } $ with a remainder of $ \color{red}{ 1418 } $.