Find remainder when $ -24x^{3}-33x^{2}+20x+3 $ is divided by $ x-5 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}5&-24&-33&20&3\\& & -120& -765& \color{black}{-3725} \\ \hline &\color{blue}{-24}&\color{blue}{-153}&\color{blue}{-745}&\color{orangered}{-3722} \end{array} $$The remainder when $ -24x^{3}-33x^{2}+20x+3 $ is divided by $ x-5 $ is $ \, \color{red}{ -3722 } $.
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 -5 = 0 $ ( $ x = \color{blue}{ 5 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{5}&-24&-33&20&3\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}5&\color{orangered}{ -24 }&-33&20&3\\& & & & \\ \hline &\color{orangered}{-24}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -24 \right) } = \color{blue}{ -120 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&-24&-33&20&3\\& & \color{blue}{-120} & & \\ \hline &\color{blue}{-24}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -33 } + \color{orangered}{ \left( -120 \right) } = \color{orangered}{ -153 } $
$$ \begin{array}{c|rrrr}5&-24&\color{orangered}{ -33 }&20&3\\& & \color{orangered}{-120} & & \\ \hline &-24&\color{orangered}{-153}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -153 \right) } = \color{blue}{ -765 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&-24&-33&20&3\\& & -120& \color{blue}{-765} & \\ \hline &-24&\color{blue}{-153}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 20 } + \color{orangered}{ \left( -765 \right) } = \color{orangered}{ -745 } $
$$ \begin{array}{c|rrrr}5&-24&-33&\color{orangered}{ 20 }&3\\& & -120& \color{orangered}{-765} & \\ \hline &-24&-153&\color{orangered}{-745}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -745 \right) } = \color{blue}{ -3725 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&-24&-33&20&3\\& & -120& -765& \color{blue}{-3725} \\ \hline &-24&-153&\color{blue}{-745}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ \left( -3725 \right) } = \color{orangered}{ -3722 } $
$$ \begin{array}{c|rrrr}5&-24&-33&20&\color{orangered}{ 3 }\\& & -120& -765& \color{orangered}{-3725} \\ \hline &\color{blue}{-24}&\color{blue}{-153}&\color{blue}{-745}&\color{orangered}{-3722} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -3722 }\right) $.