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