Divide using synthetic division $$ ( 3x^{6}+32x^{3}+64 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}-2&3&0&0&32&0&0&64\\& & -6& 12& -24& -16& 32& \color{black}{-64} \\ \hline &\color{blue}{3}&\color{blue}{-6}&\color{blue}{12}&\color{blue}{8}&\color{blue}{-16}&\color{blue}{32}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 3x^{6}+32x^{3}+64 }{ x+2 } = \color{blue}{3x^{5}-6x^{4}+12x^{3}+8x^{2}-16x+32} $$Explanation
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.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&3&0&0&32&0&0&64\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}-2&\color{orangered}{ 3 }&0&0&32&0&0&64\\& & & & & & & \\ \hline &\color{orangered}{3}&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 3 } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&3&0&0&32&0&0&64\\& & \color{blue}{-6} & & & & & \\ \hline &\color{blue}{3}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrrrrr}-2&3&\color{orangered}{ 0 }&0&32&0&0&64\\& & \color{orangered}{-6} & & & & & \\ \hline &3&\color{orangered}{-6}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ 12 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&3&0&0&32&0&0&64\\& & -6& \color{blue}{12} & & & & \\ \hline &3&\color{blue}{-6}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 12 } = \color{orangered}{ 12 } $
$$ \begin{array}{c|rrrrrrr}-2&3&0&\color{orangered}{ 0 }&32&0&0&64\\& & -6& \color{orangered}{12} & & & & \\ \hline &3&-6&\color{orangered}{12}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 12 } = \color{blue}{ -24 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&3&0&0&32&0&0&64\\& & -6& 12& \color{blue}{-24} & & & \\ \hline &3&-6&\color{blue}{12}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 32 } + \color{orangered}{ \left( -24 \right) } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrrrrr}-2&3&0&0&\color{orangered}{ 32 }&0&0&64\\& & -6& 12& \color{orangered}{-24} & & & \\ \hline &3&-6&12&\color{orangered}{8}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 8 } = \color{blue}{ -16 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&3&0&0&32&0&0&64\\& & -6& 12& -24& \color{blue}{-16} & & \\ \hline &3&-6&12&\color{blue}{8}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -16 \right) } = \color{orangered}{ -16 } $
$$ \begin{array}{c|rrrrrrr}-2&3&0&0&32&\color{orangered}{ 0 }&0&64\\& & -6& 12& -24& \color{orangered}{-16} & & \\ \hline &3&-6&12&8&\color{orangered}{-16}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -16 \right) } = \color{blue}{ 32 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&3&0&0&32&0&0&64\\& & -6& 12& -24& -16& \color{blue}{32} & \\ \hline &3&-6&12&8&\color{blue}{-16}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 32 } = \color{orangered}{ 32 } $
$$ \begin{array}{c|rrrrrrr}-2&3&0&0&32&0&\color{orangered}{ 0 }&64\\& & -6& 12& -24& -16& \color{orangered}{32} & \\ \hline &3&-6&12&8&-16&\color{orangered}{32}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 32 } = \color{blue}{ -64 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&3&0&0&32&0&0&64\\& & -6& 12& -24& -16& 32& \color{blue}{-64} \\ \hline &3&-6&12&8&-16&\color{blue}{32}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 64 } + \color{orangered}{ \left( -64 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrrr}-2&3&0&0&32&0&0&\color{orangered}{ 64 }\\& & -6& 12& -24& -16& 32& \color{orangered}{-64} \\ \hline &\color{blue}{3}&\color{blue}{-6}&\color{blue}{12}&\color{blue}{8}&\color{blue}{-16}&\color{blue}{32}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{5}-6x^{4}+12x^{3}+8x^{2}-16x+32 } $ with a remainder of $ \color{red}{ 0 } $.