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Vieta’s Formulas.md

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+#Math #Algebra
+
+Let polynomial $a$ be:
+
+$$
+a = c_n \prod _{i = 0}^n (x - r_i)
+$$
+
+where $r_i$ is a root of $a$, and $c_n$ is the leading coefficient of $a$.
+
+We can also represent $a$ as:
+
+$$
+a = \sum _{i = 0}^n c_i x^i
+$$
+
+By expanding the first definition of $a$, we can define $c_i$ by:
+
+$$
+c_{n-i} = (-1)^i c_n\sum _{sym}^i r
+$$
+
+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 $(-1)^i$
+
+We can refactor to state:
+
+$$
+\sum _{sym}^i r = (-1)^i \frac {c_{n-i}} {c_n}
+$$