Find remainder when $ x^{3}-6x^{2}+11x-6 $ is divided by $ x+2 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-2&1&-6&11&-6\\& & -2& 16& \color{black}{-54} \\ \hline &\color{blue}{1}&\color{blue}{-8}&\color{blue}{27}&\color{orangered}{-60} \end{array} $$The remainder when $ x^{3}-6x^{2}+11x-6 $ is divided by $ x+2 $ is $ \, \color{red}{ -60 } $.
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 + 2 = 0 $ ( $ x = \color{blue}{ -2 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&1&-6&11&-6\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-2&\color{orangered}{ 1 }&-6&11&-6\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 1 } = \color{blue}{ -2 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&1&-6&11&-6\\& & \color{blue}{-2} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrrr}-2&1&\color{orangered}{ -6 }&11&-6\\& & \color{orangered}{-2} & & \\ \hline &1&\color{orangered}{-8}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ 16 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&1&-6&11&-6\\& & -2& \color{blue}{16} & \\ \hline &1&\color{blue}{-8}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 11 } + \color{orangered}{ 16 } = \color{orangered}{ 27 } $
$$ \begin{array}{c|rrrr}-2&1&-6&\color{orangered}{ 11 }&-6\\& & -2& \color{orangered}{16} & \\ \hline &1&-8&\color{orangered}{27}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 27 } = \color{blue}{ -54 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&1&-6&11&-6\\& & -2& 16& \color{blue}{-54} \\ \hline &1&-8&\color{blue}{27}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ \left( -54 \right) } = \color{orangered}{ -60 } $
$$ \begin{array}{c|rrrr}-2&1&-6&11&\color{orangered}{ -6 }\\& & -2& 16& \color{orangered}{-54} \\ \hline &\color{blue}{1}&\color{blue}{-8}&\color{blue}{27}&\color{orangered}{-60} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -60 }\right) $.