Find remainder when $ 3x^{3}+7x^{2}-21x-30 $ is divided by $ x+4 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-4&3&7&-21&-30\\& & -12& 20& \color{black}{4} \\ \hline &\color{blue}{3}&\color{blue}{-5}&\color{blue}{-1}&\color{orangered}{-26} \end{array} $$The remainder when $ 3x^{3}+7x^{2}-21x-30 $ is divided by $ x+4 $ is $ \, \color{red}{ -26 } $.
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 + 4 = 0 $ ( $ x = \color{blue}{ -4 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&3&7&-21&-30\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-4&\color{orangered}{ 3 }&7&-21&-30\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 3 } = \color{blue}{ -12 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&3&7&-21&-30\\& & \color{blue}{-12} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 7 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrrr}-4&3&\color{orangered}{ 7 }&-21&-30\\& & \color{orangered}{-12} & & \\ \hline &3&\color{orangered}{-5}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ 20 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&3&7&-21&-30\\& & -12& \color{blue}{20} & \\ \hline &3&\color{blue}{-5}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -21 } + \color{orangered}{ 20 } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrr}-4&3&7&\color{orangered}{ -21 }&-30\\& & -12& \color{orangered}{20} & \\ \hline &3&-5&\color{orangered}{-1}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 4 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&3&7&-21&-30\\& & -12& 20& \color{blue}{4} \\ \hline &3&-5&\color{blue}{-1}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -30 } + \color{orangered}{ 4 } = \color{orangered}{ -26 } $
$$ \begin{array}{c|rrrr}-4&3&7&-21&\color{orangered}{ -30 }\\& & -12& 20& \color{orangered}{4} \\ \hline &\color{blue}{3}&\color{blue}{-5}&\color{blue}{-1}&\color{orangered}{-26} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -26 }\right) $.