Question
Divide using synthetic division $$ ( -8x ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rr}3&-8&0\\& & \color{black}{-24} \\ \hline &\color{blue}{-8}&\color{orangered}{-24} \end{array} $$The solution is:
$$ \dfrac{ -8x }{ x-3 } = \color{blue}{-8} \color{red}{~-~} \dfrac{ \color{red}{ 24 } }{ x-3 } $$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.
$$ \begin{array}{c|rr}\color{blue}{3}&-8&0\\& & \\ \hline && \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rr}3&\color{orangered}{ -8 }&0\\& & \\ \hline &\color{orangered}{-8}& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ -24 } $.
$$ \begin{array}{c|rr}\color{blue}{3}&-8&0\\& & \color{blue}{-24} \\ \hline &\color{blue}{-8}& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -24 \right) } = \color{orangered}{ -24 } $
$$ \begin{array}{c|rr}3&-8&\color{orangered}{ 0 }\\& & \color{orangered}{-24} \\ \hline &\color{blue}{-8}&\color{orangered}{-24} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -8 } $ with a remainder of $ \color{red}{ -24 } $.
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