Find remainder when $ 27x^{3}-125 $ is divided by $ 3x+5 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-5&27&0&0&-125\\& & -135& 675& \color{black}{-3375} \\ \hline &\color{blue}{27}&\color{blue}{-135}&\color{blue}{675}&\color{orangered}{-3500} \end{array} $$The remainder when $ 27x^{3}-125 $ is divided by $ x+5 $ is $ \, \color{red}{ -3500 } $.
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 + 5 = 0 $ ( $ x = \color{blue}{ -5 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&27&0&0&-125\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-5&\color{orangered}{ 27 }&0&0&-125\\& & & & \\ \hline &\color{orangered}{27}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 27 } = \color{blue}{ -135 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&27&0&0&-125\\& & \color{blue}{-135} & & \\ \hline &\color{blue}{27}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -135 \right) } = \color{orangered}{ -135 } $
$$ \begin{array}{c|rrrr}-5&27&\color{orangered}{ 0 }&0&-125\\& & \color{orangered}{-135} & & \\ \hline &27&\color{orangered}{-135}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -135 \right) } = \color{blue}{ 675 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&27&0&0&-125\\& & -135& \color{blue}{675} & \\ \hline &27&\color{blue}{-135}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 675 } = \color{orangered}{ 675 } $
$$ \begin{array}{c|rrrr}-5&27&0&\color{orangered}{ 0 }&-125\\& & -135& \color{orangered}{675} & \\ \hline &27&-135&\color{orangered}{675}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 675 } = \color{blue}{ -3375 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&27&0&0&-125\\& & -135& 675& \color{blue}{-3375} \\ \hline &27&-135&\color{blue}{675}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -125 } + \color{orangered}{ \left( -3375 \right) } = \color{orangered}{ -3500 } $
$$ \begin{array}{c|rrrr}-5&27&0&0&\color{orangered}{ -125 }\\& & -135& 675& \color{orangered}{-3375} \\ \hline &\color{blue}{27}&\color{blue}{-135}&\color{blue}{675}&\color{orangered}{-3500} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -3500 }\right) $.