Find remainder when $ 45x^{2}-47x-29 $ is divided by $ 3 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}0&45&-47&-29\\& & 0& \color{black}{0} \\ \hline &\color{blue}{45}&\color{blue}{-47}&\color{orangered}{-29} \end{array} $$The remainder when $ 45x^{2}-47x-29 $ is divided by $ x $ is $ \, \color{red}{ -29 } $.
Explanation
We can find remainder using synthetic division method.
Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.
$$ \begin{array}{c|rrr}\color{blue}{0}&45&-47&-29\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}0&\color{orangered}{ 45 }&-47&-29\\& & & \\ \hline &\color{orangered}{45}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 45 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrr}\color{blue}{0}&45&-47&-29\\& & \color{blue}{0} & \\ \hline &\color{blue}{45}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -47 } + \color{orangered}{ 0 } = \color{orangered}{ -47 } $
$$ \begin{array}{c|rrr}0&45&\color{orangered}{ -47 }&-29\\& & \color{orangered}{0} & \\ \hline &45&\color{orangered}{-47}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -47 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrr}\color{blue}{0}&45&-47&-29\\& & 0& \color{blue}{0} \\ \hline &45&\color{blue}{-47}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -29 } + \color{orangered}{ 0 } = \color{orangered}{ -29 } $
$$ \begin{array}{c|rrr}0&45&-47&\color{orangered}{ -29 }\\& & 0& \color{orangered}{0} \\ \hline &\color{blue}{45}&\color{blue}{-47}&\color{orangered}{-29} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -29 }\right) $.