Question
Find remainder when $ 10x $ is divided by $ x-3 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rr}3&10&0\\& & \color{black}{30} \\ \hline &\color{blue}{10}&\color{orangered}{30} \end{array} $$The remainder when $ 10x $ is divided by $ x-3 $ is $ \, \color{red}{ 30 } $.
Explanation
We can find remainder using synthetic division method.
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.
$$ \begin{array}{c|rr}\color{blue}{3}&10&0\\& & \\ \hline && \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rr}3&\color{orangered}{ 10 }&0\\& & \\ \hline &\color{orangered}{10}& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 10 } = \color{blue}{ 30 } $.
$$ \begin{array}{c|rr}\color{blue}{3}&10&0\\& & \color{blue}{30} \\ \hline &\color{blue}{10}& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 30 } = \color{orangered}{ 30 } $
$$ \begin{array}{c|rr}3&10&\color{orangered}{ 0 }\\& & \color{orangered}{30} \\ \hline &\color{blue}{10}&\color{orangered}{30} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 30 }\right) $.
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