Can somebody already acquainted with the subject explain in detail how to extract the $n$-th root of a polynomial using the method of detached coefficients? I have tried the whole day to understand books on the subject and they always end up being vague when the order of the root becomes high.
I think I have understood how to extract the square root of any polynomial and the concept of detached coefficients but I still have problems to understand what to do with the so-called columns on the left when the order of the root is large (3+) after finding the first two coefficients in the root-polynomial... I have no idea what to do recursively in the columns because all of the books I read usually explain the first steps to find the first or second coefficient of the root polynomial, and then pretty much say "etc.", but the "etc." is not precise enough for me. The books are old too, and the wording does not allow me to extrapolate and be autonomous...
I ask this here in case somebody knows, since it has been taught neither to me nor to anyone I know. These seem to be "ancient" maths and not used a lot nowadays...
Ok as requested I am providing an example of said book. The goal is to extract a cube root of order $3$ of $a^6-6a^5+21a^4-44a^3+63a^2-54a+27$
In this picture I understand the first three lines.
The $1\quad1\quad1$ makes sense to me, this is just the first coefficient repeated three times as prescribed by the usual method.
Then, the two lines that follow are just the algorithm as described
After that, I don't understand what I should do next, once I have the first three lines and found the two coefficients $1$ and $-2$.
Why $1$ in the fourth line, second column ? Why is it then changed into $-2$ ? Why are there two occurences of $-2$ in this column in the following lines instead of just one before proceeding to using the next coefficient $3$ instead?
First step in discussing factoring your sextic: gcd with derivative
$$ \left( x^{6} - 6 x^{5} + 21 x^{4} - 44 x^{3} + 63 x^{2} - 54 x + 27 \right) $$
$$ \left( x^{5} - 5 x^{4} + 14 x^{3} - 22 x^{2} + 21 x - 9 \right) $$
$$ \left( x^{6} - 6 x^{5} + 21 x^{4} - 44 x^{3} + 63 x^{2} - 54 x + 27 \right) = \left( x^{5} - 5 x^{4} + 14 x^{3} - 22 x^{2} + 21 x - 9 \right) \cdot \color{magenta}{ \left( x - 1 \right) } + \left( 2 x^{4} - 8 x^{3} + 20 x^{2} - 24 x + 18 \right) $$ $$ \left( x^{5} - 5 x^{4} + 14 x^{3} - 22 x^{2} + 21 x - 9 \right) = \left( 2 x^{4} - 8 x^{3} + 20 x^{2} - 24 x + 18 \right) \cdot \color{magenta}{ \left( \frac{ x - 1 }{ 2 } \right) } + \left( 0 \right) $$ $$ \frac{ 0}{1} $$ $$ \frac{ 1}{0} $$ $$ \color{magenta}{ \left( x - 1 \right) } \Longrightarrow \Longrightarrow \frac{ \left( x - 1 \right) }{ \left( 1 \right) } $$ $$ \color{magenta}{ \left( \frac{ x - 1 }{ 2 } \right) } \Longrightarrow \Longrightarrow \frac{ \left( \frac{ x^{2} - 2 x + 3 }{ 2 } \right) }{ \left( \frac{ x - 1 }{ 2 } \right) } $$ $$ \left( x^{2} - 2 x + 3 \right) \left( \frac{ 1}{2 } \right) - \left( x - 1 \right) \left( \frac{ x - 1 }{ 2 } \right) = \left( 1 \right) $$ $$ \left( x^{6} - 6 x^{5} + 21 x^{4} - 44 x^{3} + 63 x^{2} - 54 x + 27 \right) = \left( x^{2} - 2 x + 3 \right) \cdot \color{magenta}{ \left( x^{4} - 4 x^{3} + 10 x^{2} - 12 x + 9 \right) } + \left( 0 \right) $$ $$ \left( x^{5} - 5 x^{4} + 14 x^{3} - 22 x^{2} + 21 x - 9 \right) = \left( x - 1 \right) \cdot \color{magenta}{ \left( x^{4} - 4 x^{3} + 10 x^{2} - 12 x + 9 \right) } + \left( 0 \right) $$ $$ \mbox{GCD} = \color{magenta}{ \left( x^{4} - 4 x^{3} + 10 x^{2} - 12 x + 9 \right) } $$ $$ \left( x^{6} - 6 x^{5} + 21 x^{4} - 44 x^{3} + 63 x^{2} - 54 x + 27 \right) \left( \frac{ 1}{2 } \right) - \left( x^{5} - 5 x^{4} + 14 x^{3} - 22 x^{2} + 21 x - 9 \right) \left( \frac{ x - 1 }{ 2 } \right) = \left( x^{4} - 4 x^{3} + 10 x^{2} - 12 x + 9 \right) $$
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Pretty impressive. Let's try gcd with second derivative
$$ \left( x^{6} - 6 x^{5} + 21 x^{4} - 44 x^{3} + 63 x^{2} - 54 x + 27 \right) $$
$$ \left( 5 x^{4} - 20 x^{3} + 42 x^{2} - 44 x + 21 \right) $$
$$ \left( x^{6} - 6 x^{5} + 21 x^{4} - 44 x^{3} + 63 x^{2} - 54 x + 27 \right) = \left( 5 x^{4} - 20 x^{3} + 42 x^{2} - 44 x + 21 \right) \cdot \color{magenta}{ \left( \frac{ 5 x^{2} - 10 x + 23 }{ 25 } \right) } + \left( \frac{ 64 x^{2} - 128 x + 192 }{ 25 } \right) $$ $$ \left( 5 x^{4} - 20 x^{3} + 42 x^{2} - 44 x + 21 \right) = \left( \frac{ 64 x^{2} - 128 x + 192 }{ 25 } \right) \cdot \color{magenta}{ \left( \frac{ 125 x^{2} - 250 x + 175 }{ 64 } \right) } + \left( 0 \right) $$ $$ \frac{ 0}{1} $$ $$ \frac{ 1}{0} $$ $$ \color{magenta}{ \left( \frac{ 5 x^{2} - 10 x + 23 }{ 25 } \right) } \Longrightarrow \Longrightarrow \frac{ \left( \frac{ 5 x^{2} - 10 x + 23 }{ 25 } \right) }{ \left( 1 \right) } $$ $$ \color{magenta}{ \left( \frac{ 125 x^{2} - 250 x + 175 }{ 64 } \right) } \Longrightarrow \Longrightarrow \frac{ \left( \frac{ 25 x^{4} - 100 x^{3} + 250 x^{2} - 300 x + 225 }{ 64 } \right) }{ \left( \frac{ 125 x^{2} - 250 x + 175 }{ 64 } \right) } $$ $$ \left( x^{4} - 4 x^{3} + 10 x^{2} - 12 x + 9 \right) \left( \frac{ 25}{64 } \right) - \left( 5 x^{2} - 10 x + 7 \right) \left( \frac{ 5 x^{2} - 10 x + 23 }{ 64 } \right) = \left( 1 \right) $$ $$ \left( x^{6} - 6 x^{5} + 21 x^{4} - 44 x^{3} + 63 x^{2} - 54 x + 27 \right) = \left( x^{4} - 4 x^{3} + 10 x^{2} - 12 x + 9 \right) \cdot \color{magenta}{ \left( x^{2} - 2 x + 3 \right) } + \left( 0 \right) $$ $$ \left( 5 x^{4} - 20 x^{3} + 42 x^{2} - 44 x + 21 \right) = \left( 5 x^{2} - 10 x + 7 \right) \cdot \color{magenta}{ \left( x^{2} - 2 x + 3 \right) } + \left( 0 \right) $$ $$ \mbox{GCD} = \color{magenta}{ \left( x^{2} - 2 x + 3 \right) } $$ $$ \left( x^{6} - 6 x^{5} + 21 x^{4} - 44 x^{3} + 63 x^{2} - 54 x + 27 \right) \left( \frac{ 25}{64 } \right) - \left( 5 x^{4} - 20 x^{3} + 42 x^{2} - 44 x + 21 \right) \left( \frac{ 5 x^{2} - 10 x + 23 }{ 64 } \right) = \left( x^{2} - 2 x + 3 \right) $$