Find remainder when $ 10x^{3}+5x^{2}+75x-40 $ is divided by $ 2x+1 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-1&10&5&75&-40\\& & -10& 5& \color{black}{-80} \\ \hline &\color{blue}{10}&\color{blue}{-5}&\color{blue}{80}&\color{orangered}{-120} \end{array} $$The remainder when $ 10x^{3}+5x^{2}+75x-40 $ is divided by $ x+1 $ is $ \, \color{red}{ -120 } $.
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.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&10&5&75&-40\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-1&\color{orangered}{ 10 }&5&75&-40\\& & & & \\ \hline &\color{orangered}{10}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 10 } = \color{blue}{ -10 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&10&5&75&-40\\& & \color{blue}{-10} & & \\ \hline &\color{blue}{10}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ \left( -10 \right) } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrrr}-1&10&\color{orangered}{ 5 }&75&-40\\& & \color{orangered}{-10} & & \\ \hline &10&\color{orangered}{-5}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ 5 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&10&5&75&-40\\& & -10& \color{blue}{5} & \\ \hline &10&\color{blue}{-5}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 75 } + \color{orangered}{ 5 } = \color{orangered}{ 80 } $
$$ \begin{array}{c|rrrr}-1&10&5&\color{orangered}{ 75 }&-40\\& & -10& \color{orangered}{5} & \\ \hline &10&-5&\color{orangered}{80}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 80 } = \color{blue}{ -80 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&10&5&75&-40\\& & -10& 5& \color{blue}{-80} \\ \hline &10&-5&\color{blue}{80}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -40 } + \color{orangered}{ \left( -80 \right) } = \color{orangered}{ -120 } $
$$ \begin{array}{c|rrrr}-1&10&5&75&\color{orangered}{ -40 }\\& & -10& 5& \color{orangered}{-80} \\ \hline &\color{blue}{10}&\color{blue}{-5}&\color{blue}{80}&\color{orangered}{-120} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -120 }\right) $.