Difference between revisions of "Vieta's Formulas"
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| − | + | == Background == | |
| + | |||
Let <math>P(x)={a_n}x^n+{a_{n-1}}x^{n-1}+\cdots+{a_1}x+a_0</math>, | Let <math>P(x)={a_n}x^n+{a_{n-1}}x^{n-1}+\cdots+{a_1}x+a_0</math>, | ||
| − | where the coefficient of <math>x^{i}</math> is <math>{a}_i</math>. As a consequence of the [[Fundamental Theorem of Algebra]], we can also write | + | where the coefficient of <math>x^{i}</math> is <math>{a}_i</math>. As a consequence of the [[Fundamental Theorem of Algebra]], we can also write <math>P(x)=a_n(x-r_1)(x-r_2)\cdots(x-r_n)</math>, where <math>{r}_i</math> are the roots of <math>P(x)</math>. |
| − | |||
| − | where <math>{r}_i</math> are the roots of <math>P(x)</math>. | ||
| − | + | Also, let <math>{\sigma}_k</math> be the <math>{}{k}</math>th [[symmetric sum]]. | |
| − | + | == Statement == | |
<math>\sigma_k = (-1)^k\cdot \frac{a_{n-k}}{a_n{}}</math>, for <math>{}1\le k\le {n}</math>. | <math>\sigma_k = (-1)^k\cdot \frac{a_{n-k}}{a_n{}}</math>, for <math>{}1\le k\le {n}</math>. | ||
| − | + | == Proof == | |
[needs to be added] | [needs to be added] | ||
Revision as of 15:50, 18 June 2006
Background
Let 
,
where the coefficient of 
is 
. As a consequence of the Fundamental Theorem of Algebra, we can also write 
, where 
are the roots of 
.
Also, let 
be the 
th symmetric sum.
Statement
, for 
.
Proof
[needs to be added]
