Question
Divide using synthetic division $$ ( -5x+4 ) \div ( x+1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rr}-1&-5&4\\& & \color{black}{5} \\ \hline &\color{blue}{-5}&\color{orangered}{9} \end{array} $$The solution is:
$$ \dfrac{ -5x+4 }{ x+1 } = \color{blue}{-5} ~+~ \dfrac{ \color{red}{ 9 } }{ x+1 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 1 = 0 $ ( $ x = \color{blue}{ -1 } $ ) at the left.
$$ \begin{array}{c|rr}\color{blue}{-1}&-5&4\\& & \\ \hline && \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rr}-1&\color{orangered}{ -5 }&4\\& & \\ \hline &\color{orangered}{-5}& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ 5 } $.
$$ \begin{array}{c|rr}\color{blue}{-1}&-5&4\\& & \color{blue}{5} \\ \hline &\color{blue}{-5}& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 5 } = \color{orangered}{ 9 } $
$$ \begin{array}{c|rr}-1&-5&\color{orangered}{ 4 }\\& & \color{orangered}{5} \\ \hline &\color{blue}{-5}&\color{orangered}{9} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -5 } $ with a remainder of $ \color{red}{ 9 } $.
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