Find remainder when $ -8x^{3}-3x^{2}+8x+1035 $ is divided by $ x-5 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}5&-8&-3&8&1035\\& & -40& -215& \color{black}{-1035} \\ \hline &\color{blue}{-8}&\color{blue}{-43}&\color{blue}{-207}&\color{orangered}{0} \end{array} $$The remainder when $ -8x^{3}-3x^{2}+8x+1035 $ is divided by $ x-5 $ 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 -5 = 0 $ ( $ x = \color{blue}{ 5 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{5}&-8&-3&8&1035\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}5&\color{orangered}{ -8 }&-3&8&1035\\& & & & \\ \hline &\color{orangered}{-8}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ -40 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&-8&-3&8&1035\\& & \color{blue}{-40} & & \\ \hline &\color{blue}{-8}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ \left( -40 \right) } = \color{orangered}{ -43 } $
$$ \begin{array}{c|rrrr}5&-8&\color{orangered}{ -3 }&8&1035\\& & \color{orangered}{-40} & & \\ \hline &-8&\color{orangered}{-43}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -43 \right) } = \color{blue}{ -215 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&-8&-3&8&1035\\& & -40& \color{blue}{-215} & \\ \hline &-8&\color{blue}{-43}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ \left( -215 \right) } = \color{orangered}{ -207 } $
$$ \begin{array}{c|rrrr}5&-8&-3&\color{orangered}{ 8 }&1035\\& & -40& \color{orangered}{-215} & \\ \hline &-8&-43&\color{orangered}{-207}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -207 \right) } = \color{blue}{ -1035 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&-8&-3&8&1035\\& & -40& -215& \color{blue}{-1035} \\ \hline &-8&-43&\color{blue}{-207}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 1035 } + \color{orangered}{ \left( -1035 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}5&-8&-3&8&\color{orangered}{ 1035 }\\& & -40& -215& \color{orangered}{-1035} \\ \hline &\color{blue}{-8}&\color{blue}{-43}&\color{blue}{-207}&\color{orangered}{0} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 0 }\right) $.