29 lines
691 B
Markdown
29 lines
691 B
Markdown
#Math #Algebra
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Let polynomial $a$ be:
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$$
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a = c_n \prod _{i = 0}^n (x - r_i)
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$$
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where $r_i$ is a root of $a$, and $c_n$ is the leading coefficient of $a$.
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We can also represent $a$ as:
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$$
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a = \sum _{i = 0}^n c_i x^i
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$$
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By expanding the first definition of $a$, we can define $c_i$ by:
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$$
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c_{n-i} = (-1)^i c_n\sum _{sym}^i r
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$$
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This is through the nature of multiplying binomials, with the coefficient resulting in the sum of all possible combinations of $i$ roots multiplied together, or the $i$th elementary symmetric sum of set $r$. We also have to multiply by the negative sign, resulting in $(-1)^i$
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We can refactor to state:
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$$
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\sum _{sym}^i r = (-1)^i \frac {c_{n-i}} {c_n}
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$$ |