Divide using synthetic division $$ ( x^{7}-128 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrrr}2&1&0&0&0&0&0&0&-128\\& & 2& 4& 8& 16& 32& 64& \color{black}{128} \\ \hline &\color{blue}{1}&\color{blue}{2}&\color{blue}{4}&\color{blue}{8}&\color{blue}{16}&\color{blue}{32}&\color{blue}{64}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{7}-128 }{ x-2 } = \color{blue}{x^{6}+2x^{5}+4x^{4}+8x^{3}+16x^{2}+32x+64} $$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.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{2}&1&0&0&0&0&0&0&-128\\& & & & & & & & \\ \hline &&&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrrr}2&\color{orangered}{ 1 }&0&0&0&0&0&0&-128\\& & & & & & & & \\ \hline &\color{orangered}{1}&&&&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{2}&1&0&0&0&0&0&0&-128\\& & \color{blue}{2} & & & & & & \\ \hline &\color{blue}{1}&&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 2 } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrrrrrr}2&1&\color{orangered}{ 0 }&0&0&0&0&0&-128\\& & \color{orangered}{2} & & & & & & \\ \hline &1&\color{orangered}{2}&&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 2 } = \color{blue}{ 4 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{2}&1&0&0&0&0&0&0&-128\\& & 2& \color{blue}{4} & & & & & \\ \hline &1&\color{blue}{2}&&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 4 } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrrrrrr}2&1&0&\color{orangered}{ 0 }&0&0&0&0&-128\\& & 2& \color{orangered}{4} & & & & & \\ \hline &1&2&\color{orangered}{4}&&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 4 } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{2}&1&0&0&0&0&0&0&-128\\& & 2& 4& \color{blue}{8} & & & & \\ \hline &1&2&\color{blue}{4}&&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 8 } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrrrrrr}2&1&0&0&\color{orangered}{ 0 }&0&0&0&-128\\& & 2& 4& \color{orangered}{8} & & & & \\ \hline &1&2&4&\color{orangered}{8}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{2}&1&0&0&0&0&0&0&-128\\& & 2& 4& 8& \color{blue}{16} & & & \\ \hline &1&2&4&\color{blue}{8}&&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 16 } = \color{orangered}{ 16 } $
$$ \begin{array}{c|rrrrrrrr}2&1&0&0&0&\color{orangered}{ 0 }&0&0&-128\\& & 2& 4& 8& \color{orangered}{16} & & & \\ \hline &1&2&4&8&\color{orangered}{16}&&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 16 } = \color{blue}{ 32 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{2}&1&0&0&0&0&0&0&-128\\& & 2& 4& 8& 16& \color{blue}{32} & & \\ \hline &1&2&4&8&\color{blue}{16}&&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 32 } = \color{orangered}{ 32 } $
$$ \begin{array}{c|rrrrrrrr}2&1&0&0&0&0&\color{orangered}{ 0 }&0&-128\\& & 2& 4& 8& 16& \color{orangered}{32} & & \\ \hline &1&2&4&8&16&\color{orangered}{32}&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{2}&1&0&0&0&0&0&0&-128\\& & 2& 4& 8& 16& 32& \color{blue}{64} & \\ \hline &1&2&4&8&16&\color{blue}{32}&& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 64 } = \color{orangered}{ 64 } $
$$ \begin{array}{c|rrrrrrrr}2&1&0&0&0&0&0&\color{orangered}{ 0 }&-128\\& & 2& 4& 8& 16& 32& \color{orangered}{64} & \\ \hline &1&2&4&8&16&32&\color{orangered}{64}& \end{array} $$Step 14 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 64 } = \color{blue}{ 128 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{2}&1&0&0&0&0&0&0&-128\\& & 2& 4& 8& 16& 32& 64& \color{blue}{128} \\ \hline &1&2&4&8&16&32&\color{blue}{64}& \end{array} $$Step 15 : Add down last column: $ \color{orangered}{ -128 } + \color{orangered}{ 128 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrrrr}2&1&0&0&0&0&0&0&\color{orangered}{ -128 }\\& & 2& 4& 8& 16& 32& 64& \color{orangered}{128} \\ \hline &\color{blue}{1}&\color{blue}{2}&\color{blue}{4}&\color{blue}{8}&\color{blue}{16}&\color{blue}{32}&\color{blue}{64}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{6}+2x^{5}+4x^{4}+8x^{3}+16x^{2}+32x+64 } $ with a remainder of $ \color{red}{ 0 } $.