Find remainder when $ 3x^{3}+9x^{2}-36x-30 $ is divided by $ x+5 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-5&3&9&-36&-30\\& & -15& 30& \color{black}{30} \\ \hline &\color{blue}{3}&\color{blue}{-6}&\color{blue}{-6}&\color{orangered}{0} \end{array} $$The remainder when $ 3x^{3}+9x^{2}-36x-30 $ is divided by $ x+5 $ is $ \, \color{red}{ 0 } $.
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.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&3&9&-36&-30\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-5&\color{orangered}{ 3 }&9&-36&-30\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 3 } = \color{blue}{ -15 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&3&9&-36&-30\\& & \color{blue}{-15} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ \left( -15 \right) } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrr}-5&3&\color{orangered}{ 9 }&-36&-30\\& & \color{orangered}{-15} & & \\ \hline &3&\color{orangered}{-6}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ 30 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&3&9&-36&-30\\& & -15& \color{blue}{30} & \\ \hline &3&\color{blue}{-6}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -36 } + \color{orangered}{ 30 } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrr}-5&3&9&\color{orangered}{ -36 }&-30\\& & -15& \color{orangered}{30} & \\ \hline &3&-6&\color{orangered}{-6}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ 30 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&3&9&-36&-30\\& & -15& 30& \color{blue}{30} \\ \hline &3&-6&\color{blue}{-6}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -30 } + \color{orangered}{ 30 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}-5&3&9&-36&\color{orangered}{ -30 }\\& & -15& 30& \color{orangered}{30} \\ \hline &\color{blue}{3}&\color{blue}{-6}&\color{blue}{-6}&\color{orangered}{0} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 0 }\right) $.