Find remainder when $ 5x^{4}-3x^{3}+2023 $ is divided by $ x+1 $.

The synthetic division table is:

$$ \begin{array}{c|rrrrr}-1&5&-3&0&0&2023\\& & -5& 8& -8& \color{black}{8} \\ \hline &\color{blue}{5}&\color{blue}{-8}&\color{blue}{8}&\color{blue}{-8}&\color{orangered}{2031} \end{array} $$

The remainder when $ 5x^{4}-3x^{3}+2023 $ is divided by $ x+1 $ is $ \, \color{red}{ 2031 } $.

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.

$$ \begin{array}{c|rrrrr}\color{blue}{-1}&5&-3&0&0&2023\\& & & & & \\ \hline &&&&& \end{array} $$

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

$$ \begin{array}{c|rrrrr}-1&\color{orangered}{ 5 }&-3&0&0&2023\\& & & & & \\ \hline &\color{orangered}{5}&&&& \end{array} $$

Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 5 } = \color{blue}{ -5 } $.

$$ \begin{array}{c|rrrrr}\color{blue}{-1}&5&-3&0&0&2023\\& & \color{blue}{-5} & & & \\ \hline &\color{blue}{5}&&&& \end{array} $$

Step 3 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ \left( -5 \right) } = \color{orangered}{ -8 } $

$$ \begin{array}{c|rrrrr}-1&5&\color{orangered}{ -3 }&0&0&2023\\& & \color{orangered}{-5} & & & \\ \hline &5&\color{orangered}{-8}&&& \end{array} $$

Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ 8 } $.

$$ \begin{array}{c|rrrrr}\color{blue}{-1}&5&-3&0&0&2023\\& & -5& \color{blue}{8} & & \\ \hline &5&\color{blue}{-8}&&& \end{array} $$

Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 8 } = \color{orangered}{ 8 } $

$$ \begin{array}{c|rrrrr}-1&5&-3&\color{orangered}{ 0 }&0&2023\\& & -5& \color{orangered}{8} & & \\ \hline &5&-8&\color{orangered}{8}&& \end{array} $$

Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 8 } = \color{blue}{ -8 } $.

$$ \begin{array}{c|rrrrr}\color{blue}{-1}&5&-3&0&0&2023\\& & -5& 8& \color{blue}{-8} & \\ \hline &5&-8&\color{blue}{8}&& \end{array} $$

Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ -8 } $

$$ \begin{array}{c|rrrrr}-1&5&-3&0&\color{orangered}{ 0 }&2023\\& & -5& 8& \color{orangered}{-8} & \\ \hline &5&-8&8&\color{orangered}{-8}& \end{array} $$

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

$$ \begin{array}{c|rrrrr}\color{blue}{-1}&5&-3&0&0&2023\\& & -5& 8& -8& \color{blue}{8} \\ \hline &5&-8&8&\color{blue}{-8}& \end{array} $$

Step 9 : Add down last column: $ \color{orangered}{ 2023 } + \color{orangered}{ 8 } = \color{orangered}{ 2031 } $

$$ \begin{array}{c|rrrrr}-1&5&-3&0&0&\color{orangered}{ 2023 }\\& & -5& 8& -8& \color{orangered}{8} \\ \hline &\color{blue}{5}&\color{blue}{-8}&\color{blue}{8}&\color{blue}{-8}&\color{orangered}{2031} \end{array} $$

Remainder is the last entry in the bottom row $ \left(\color{red}{ 2031 }\right) $.

Synthetic Division Calculator