Difference between revisions of "Vieta's Formulas"
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* [[Algebra]] | * [[Algebra]] | ||
* [[Polynomials]] | * [[Polynomials]] | ||
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== Related Links == | == Related Links == | ||
[http://mathworld.wolfram.com/VietasFormulas.html Mathworld's Article] | [http://mathworld.wolfram.com/VietasFormulas.html Mathworld's Article] | ||
Revision as of 20:37, 12 July 2006
Introduction
Let 
,
where the coefficient of 
is 
. As a consequence of the Fundamental Theorem of Algebra, we can also write 
, where 
are the roots of 
. We thus have that
Expanding out the right hand side gives us
We can see that the coefficient of 
will be the 
th symmetric sum. The th symmetric sum is just the sum of the roots taken at a time. For example, the 4th symmetric sum is 
Notice that every possible combination of four roots shows up in this sum.
We now have two different expressions for . These must be equal. However, the only way for two polynomials to be equal for all values of 
is for each of their corresponding coefficients to be equal. So, starting with the coefficient of 
, we see that
More commonly, these are written with the roots on one side and the 
on the other (this can be arrived at by dividing both sides of all the equations by 
).
If we denote 
as the th symmetric sum, then we can write those formulas more compactly as 
, for 
.
