Difference between revisions of "Vieta's Formulas"
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=== Background === | === Background === | ||
| − | Let | + | 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 | ||
| − | + | <center><math>P(x)=a_n(x-r_1)(x-r_2)\cdots(x-r_n)</math>,</center> | |
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>. | ||
| − | Let <math>{\sigma}_k</math> be the <math>{k}</math>th [[symmetric sum]]. | + | Let <math>{\sigma}_k</math> be the <math>{}{k}</math>th [[symmetric sum]]. |
=== Statement === | === Statement === | ||
| + | Vieta's says that <math>\sigma_k = (-1)^k\cdot \frac{a_{n-k}}{a_n{}}</math>, | ||
| − | + | for <math>{}1\le k\le {n}</math>. | |
| − | |||
| − | |||
=== Proof === | === Proof === | ||
[needs to be added] | [needs to be added] | ||
Revision as of 14:26, 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 
.
Let 
be the 
th symmetric sum.
Statement
Vieta's says that 
,
for 
.
Proof
[needs to be added]
