Find remainder when $ 11x^{3}+3x-4 $ is divided by $ x+4 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-4&11&0&3&-4\\& & -44& 176& \color{black}{-716} \\ \hline &\color{blue}{11}&\color{blue}{-44}&\color{blue}{179}&\color{orangered}{-720} \end{array} $$The remainder when $ 11x^{3}+3x-4 $ is divided by $ x+4 $ is $ \, \color{red}{ -720 } $.
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.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&11&0&3&-4\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-4&\color{orangered}{ 11 }&0&3&-4\\& & & & \\ \hline &\color{orangered}{11}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 11 } = \color{blue}{ -44 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&11&0&3&-4\\& & \color{blue}{-44} & & \\ \hline &\color{blue}{11}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -44 \right) } = \color{orangered}{ -44 } $
$$ \begin{array}{c|rrrr}-4&11&\color{orangered}{ 0 }&3&-4\\& & \color{orangered}{-44} & & \\ \hline &11&\color{orangered}{-44}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -44 \right) } = \color{blue}{ 176 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&11&0&3&-4\\& & -44& \color{blue}{176} & \\ \hline &11&\color{blue}{-44}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 176 } = \color{orangered}{ 179 } $
$$ \begin{array}{c|rrrr}-4&11&0&\color{orangered}{ 3 }&-4\\& & -44& \color{orangered}{176} & \\ \hline &11&-44&\color{orangered}{179}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 179 } = \color{blue}{ -716 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&11&0&3&-4\\& & -44& 176& \color{blue}{-716} \\ \hline &11&-44&\color{blue}{179}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ \left( -716 \right) } = \color{orangered}{ -720 } $
$$ \begin{array}{c|rrrr}-4&11&0&3&\color{orangered}{ -4 }\\& & -44& 176& \color{orangered}{-716} \\ \hline &\color{blue}{11}&\color{blue}{-44}&\color{blue}{179}&\color{orangered}{-720} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -720 }\right) $.