Find remainder when $ 37x^{3}+39x^{2}+22x+23 $ is divided by $ x+1 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-1&37&39&22&23\\& & -37& -2& \color{black}{-20} \\ \hline &\color{blue}{37}&\color{blue}{2}&\color{blue}{20}&\color{orangered}{3} \end{array} $$The remainder when $ 37x^{3}+39x^{2}+22x+23 $ is divided by $ x+1 $ is $ \, \color{red}{ 3 } $.
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 + 1 = 0 $ ( $ x = \color{blue}{ -1 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&37&39&22&23\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-1&\color{orangered}{ 37 }&39&22&23\\& & & & \\ \hline &\color{orangered}{37}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 37 } = \color{blue}{ -37 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&37&39&22&23\\& & \color{blue}{-37} & & \\ \hline &\color{blue}{37}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 39 } + \color{orangered}{ \left( -37 \right) } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrr}-1&37&\color{orangered}{ 39 }&22&23\\& & \color{orangered}{-37} & & \\ \hline &37&\color{orangered}{2}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 2 } = \color{blue}{ -2 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&37&39&22&23\\& & -37& \color{blue}{-2} & \\ \hline &37&\color{blue}{2}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 22 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ 20 } $
$$ \begin{array}{c|rrrr}-1&37&39&\color{orangered}{ 22 }&23\\& & -37& \color{orangered}{-2} & \\ \hline &37&2&\color{orangered}{20}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 20 } = \color{blue}{ -20 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&37&39&22&23\\& & -37& -2& \color{blue}{-20} \\ \hline &37&2&\color{blue}{20}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 23 } + \color{orangered}{ \left( -20 \right) } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrr}-1&37&39&22&\color{orangered}{ 23 }\\& & -37& -2& \color{orangered}{-20} \\ \hline &\color{blue}{37}&\color{blue}{2}&\color{blue}{20}&\color{orangered}{3} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 3 }\right) $.