Find remainder when $ 4x^{4}+4x^{3} $ is divided by $ x+1 $.

The synthetic division table is:

$$ \begin{array}{c|rrrrr}-1&4&4&0&0&0\\& & -4& 0& 0& \color{black}{0} \\ \hline &\color{blue}{4}&\color{blue}{0}&\color{blue}{0}&\color{blue}{0}&\color{orangered}{0} \end{array} $$

The remainder when $ 4x^{4}+4x^{3} $ is divided by $ x+1 $ is $ \, \color{red}{ 0 } $.

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 + 1 = 0 $ ( $ x = \color{blue}{ -1 } $ ) at the left.

$$ \begin{array}{c|rrrrr}\color{blue}{-1}&4&4&0&0&0\\& & & & & \\ \hline &&&&& \end{array} $$

Step 1 : Bring down the leading coefficient to the bottom row.

$$ \begin{array}{c|rrrrr}-1&\color{orangered}{ 4 }&4&0&0&0\\& & & & & \\ \hline &\color{orangered}{4}&&&& \end{array} $$

Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 4 } = \color{blue}{ -4 } $.

$$ \begin{array}{c|rrrrr}\color{blue}{-1}&4&4&0&0&0\\& & \color{blue}{-4} & & & \\ \hline &\color{blue}{4}&&&& \end{array} $$

Step 3 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ 0 } $

$$ \begin{array}{c|rrrrr}-1&4&\color{orangered}{ 4 }&0&0&0\\& & \color{orangered}{-4} & & & \\ \hline &4&\color{orangered}{0}&&& \end{array} $$

Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.

$$ \begin{array}{c|rrrrr}\color{blue}{-1}&4&4&0&0&0\\& & -4& \color{blue}{0} & & \\ \hline &4&\color{blue}{0}&&& \end{array} $$

Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $

$$ \begin{array}{c|rrrrr}-1&4&4&\color{orangered}{ 0 }&0&0\\& & -4& \color{orangered}{0} & & \\ \hline &4&0&\color{orangered}{0}&& \end{array} $$

Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.

$$ \begin{array}{c|rrrrr}\color{blue}{-1}&4&4&0&0&0\\& & -4& 0& \color{blue}{0} & \\ \hline &4&0&\color{blue}{0}&& \end{array} $$

Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $

$$ \begin{array}{c|rrrrr}-1&4&4&0&\color{orangered}{ 0 }&0\\& & -4& 0& \color{orangered}{0} & \\ \hline &4&0&0&\color{orangered}{0}& \end{array} $$

Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.

$$ \begin{array}{c|rrrrr}\color{blue}{-1}&4&4&0&0&0\\& & -4& 0& 0& \color{blue}{0} \\ \hline &4&0&0&\color{blue}{0}& \end{array} $$

Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $

$$ \begin{array}{c|rrrrr}-1&4&4&0&0&\color{orangered}{ 0 }\\& & -4& 0& 0& \color{orangered}{0} \\ \hline &\color{blue}{4}&\color{blue}{0}&\color{blue}{0}&\color{blue}{0}&\color{orangered}{0} \end{array} $$

Remainder is the last entry in the bottom row $ \left(\color{red}{ 0 }\right) $.

Synthetic Division Calculator