Divide using synthetic division $$ ( -7x^{2}+2x+2 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}3&-7&2&2\\& & -21& \color{black}{-57} \\ \hline &\color{blue}{-7}&\color{blue}{-19}&\color{orangered}{-55} \end{array} $$The solution is:
$$ \frac{ -7x^{2}+2x+2 }{ x-3 } = \color{blue}{-7x-19} \color{red}{~-~} \frac{ \color{red}{ 55 } }{ x-3 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -3 = 0 $ ( $ x = \color{blue}{ 3 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{3}&-7&2&2\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}3&\color{orangered}{ -7 }&2&2\\& & & \\ \hline &\color{orangered}{-7}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -7 \right) } = \color{blue}{ -21 } $.
$$ \begin{array}{c|rrr}\color{blue}{3}&-7&2&2\\& & \color{blue}{-21} & \\ \hline &\color{blue}{-7}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -21 \right) } = \color{orangered}{ -19 } $
$$ \begin{array}{c|rrr}3&-7&\color{orangered}{ 2 }&2\\& & \color{orangered}{-21} & \\ \hline &-7&\color{orangered}{-19}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -19 \right) } = \color{blue}{ -57 } $.
$$ \begin{array}{c|rrr}\color{blue}{3}&-7&2&2\\& & -21& \color{blue}{-57} \\ \hline &-7&\color{blue}{-19}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -57 \right) } = \color{orangered}{ -55 } $
$$ \begin{array}{c|rrr}3&-7&2&\color{orangered}{ 2 }\\& & -21& \color{orangered}{-57} \\ \hline &\color{blue}{-7}&\color{blue}{-19}&\color{orangered}{-55} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -7x-19 } $ with a remainder of $ \color{red}{ -55 } $.