Find remainder when $ 2x^{6}+56x^{3}+54 $ is divided by $ x+3 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}-3&2&0&0&56&0&0&54\\& & -6& 18& -54& -6& 18& \color{black}{-54} \\ \hline &\color{blue}{2}&\color{blue}{-6}&\color{blue}{18}&\color{blue}{2}&\color{blue}{-6}&\color{blue}{18}&\color{orangered}{0} \end{array} $$The remainder when $ 2x^{6}+56x^{3}+54 $ is divided by $ x+3 $ 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 + 3 = 0 $ ( $ x = \color{blue}{ -3 } $ ) at the left.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-3}&2&0&0&56&0&0&54\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}-3&\color{orangered}{ 2 }&0&0&56&0&0&54\\& & & & & & & \\ \hline &\color{orangered}{2}&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 2 } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-3}&2&0&0&56&0&0&54\\& & \color{blue}{-6} & & & & & \\ \hline &\color{blue}{2}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrrrrr}-3&2&\color{orangered}{ 0 }&0&56&0&0&54\\& & \color{orangered}{-6} & & & & & \\ \hline &2&\color{orangered}{-6}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-3}&2&0&0&56&0&0&54\\& & -6& \color{blue}{18} & & & & \\ \hline &2&\color{blue}{-6}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 18 } = \color{orangered}{ 18 } $
$$ \begin{array}{c|rrrrrrr}-3&2&0&\color{orangered}{ 0 }&56&0&0&54\\& & -6& \color{orangered}{18} & & & & \\ \hline &2&-6&\color{orangered}{18}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 18 } = \color{blue}{ -54 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-3}&2&0&0&56&0&0&54\\& & -6& 18& \color{blue}{-54} & & & \\ \hline &2&-6&\color{blue}{18}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 56 } + \color{orangered}{ \left( -54 \right) } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrrrrr}-3&2&0&0&\color{orangered}{ 56 }&0&0&54\\& & -6& 18& \color{orangered}{-54} & & & \\ \hline &2&-6&18&\color{orangered}{2}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 2 } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-3}&2&0&0&56&0&0&54\\& & -6& 18& -54& \color{blue}{-6} & & \\ \hline &2&-6&18&\color{blue}{2}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrrrrr}-3&2&0&0&56&\color{orangered}{ 0 }&0&54\\& & -6& 18& -54& \color{orangered}{-6} & & \\ \hline &2&-6&18&2&\color{orangered}{-6}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-3}&2&0&0&56&0&0&54\\& & -6& 18& -54& -6& \color{blue}{18} & \\ \hline &2&-6&18&2&\color{blue}{-6}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 18 } = \color{orangered}{ 18 } $
$$ \begin{array}{c|rrrrrrr}-3&2&0&0&56&0&\color{orangered}{ 0 }&54\\& & -6& 18& -54& -6& \color{orangered}{18} & \\ \hline &2&-6&18&2&-6&\color{orangered}{18}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 18 } = \color{blue}{ -54 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-3}&2&0&0&56&0&0&54\\& & -6& 18& -54& -6& 18& \color{blue}{-54} \\ \hline &2&-6&18&2&-6&\color{blue}{18}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 54 } + \color{orangered}{ \left( -54 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrrr}-3&2&0&0&56&0&0&\color{orangered}{ 54 }\\& & -6& 18& -54& -6& 18& \color{orangered}{-54} \\ \hline &\color{blue}{2}&\color{blue}{-6}&\color{blue}{18}&\color{blue}{2}&\color{blue}{-6}&\color{blue}{18}&\color{orangered}{0} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 0 }\right) $.