Divide using synthetic division $$ ( 8x^{5}+32x^{4}+5x ) \div ( x-4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}4&8&32&0&0&5&0\\& & 32& 256& 1024& 4096& \color{black}{16404} \\ \hline &\color{blue}{8}&\color{blue}{64}&\color{blue}{256}&\color{blue}{1024}&\color{blue}{4101}&\color{orangered}{16404} \end{array} $$The solution is:
$$ \frac{ 8x^{5}+32x^{4}+5x }{ x-4 } = \color{blue}{8x^{4}+64x^{3}+256x^{2}+1024x+4101} ~+~ \frac{ \color{red}{ 16404 } }{ x-4 } $$Explanation
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.
$$ \begin{array}{c|rrrrrr}\color{blue}{4}&8&32&0&0&5&0\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}4&\color{orangered}{ 8 }&32&0&0&5&0\\& & & & & & \\ \hline &\color{orangered}{8}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 8 } = \color{blue}{ 32 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{4}&8&32&0&0&5&0\\& & \color{blue}{32} & & & & \\ \hline &\color{blue}{8}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 32 } + \color{orangered}{ 32 } = \color{orangered}{ 64 } $
$$ \begin{array}{c|rrrrrr}4&8&\color{orangered}{ 32 }&0&0&5&0\\& & \color{orangered}{32} & & & & \\ \hline &8&\color{orangered}{64}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 64 } = \color{blue}{ 256 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{4}&8&32&0&0&5&0\\& & 32& \color{blue}{256} & & & \\ \hline &8&\color{blue}{64}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 256 } = \color{orangered}{ 256 } $
$$ \begin{array}{c|rrrrrr}4&8&32&\color{orangered}{ 0 }&0&5&0\\& & 32& \color{orangered}{256} & & & \\ \hline &8&64&\color{orangered}{256}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 256 } = \color{blue}{ 1024 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{4}&8&32&0&0&5&0\\& & 32& 256& \color{blue}{1024} & & \\ \hline &8&64&\color{blue}{256}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 1024 } = \color{orangered}{ 1024 } $
$$ \begin{array}{c|rrrrrr}4&8&32&0&\color{orangered}{ 0 }&5&0\\& & 32& 256& \color{orangered}{1024} & & \\ \hline &8&64&256&\color{orangered}{1024}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 1024 } = \color{blue}{ 4096 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{4}&8&32&0&0&5&0\\& & 32& 256& 1024& \color{blue}{4096} & \\ \hline &8&64&256&\color{blue}{1024}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 4096 } = \color{orangered}{ 4101 } $
$$ \begin{array}{c|rrrrrr}4&8&32&0&0&\color{orangered}{ 5 }&0\\& & 32& 256& 1024& \color{orangered}{4096} & \\ \hline &8&64&256&1024&\color{orangered}{4101}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 4101 } = \color{blue}{ 16404 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{4}&8&32&0&0&5&0\\& & 32& 256& 1024& 4096& \color{blue}{16404} \\ \hline &8&64&256&1024&\color{blue}{4101}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 16404 } = \color{orangered}{ 16404 } $
$$ \begin{array}{c|rrrrrr}4&8&32&0&0&5&\color{orangered}{ 0 }\\& & 32& 256& 1024& 4096& \color{orangered}{16404} \\ \hline &\color{blue}{8}&\color{blue}{64}&\color{blue}{256}&\color{blue}{1024}&\color{blue}{4101}&\color{orangered}{16404} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 8x^{4}+64x^{3}+256x^{2}+1024x+4101 } $ with a remainder of $ \color{red}{ 16404 } $.