Difference between revisions of "Vieta's Formulas"
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| + | '''Vieta's formulas''' are a set of [[equation]]s relating the [[root]]s and the [[coefficient]]s of [[polynomial]]s. | ||
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== Introduction == | == Introduction == | ||
| − | 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)</math> be a polynomial of degree <math>n</math>, so <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>\displaystyle x^{i}</math> is <math>\displaystyle {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>\displaystyle {r}_i</math> are the roots of <math>\displaystyle P(x)</math>. We thus have that | + | where the coefficient of <math>\displaystyle x^{i}</math> is <math>\displaystyle {a}_i</math> and <math>a_n \neq 0</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>\displaystyle {r}_i</math> are the roots of <math>\displaystyle P(x)</math>. We thus have that |
<math> a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0 = a_n(x-r_1)(x-r_2)\cdots(x-r_n).</math> | <math> a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0 = a_n(x-r_1)(x-r_2)\cdots(x-r_n).</math> | ||
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<center><math> a_nx^n - a_n(r_1+r_2+\cdots+r_n)x^{n-1} + a_n(r_1r_2 + r_1r_3 + \cdots + r_{n-1}r_n)x^{n-2} + \cdots + (-1)^na_n r_1r_2\cdots r_n.</math></center> | <center><math> a_nx^n - a_n(r_1+r_2+\cdots+r_n)x^{n-1} + a_n(r_1r_2 + r_1r_3 + \cdots + r_{n-1}r_n)x^{n-2} + \cdots + (-1)^na_n r_1r_2\cdots r_n.</math></center> | ||
| − | + | The coefficient of <math>\displaystyle x^k </math> in this expression will be the <math> \displaystyle k </math>th [[symmetric sum]] of the <math>r_i</math>. | |
| − | We now have two different expressions for <math>\displaystyle P(x)</math>. These | + | We now have two different expressions for <math>\displaystyle P(x)</math>. These must be equal. However, the only way for two polynomials to be equal for all values of <math>\displaystyle x</math> is for each of their corresponding coefficients to be equal. So, starting with the coefficient of <math>\displaystyle x^n </math>, we see that |
<center><math>\displaystyle a_n = a_n</math></center> | <center><math>\displaystyle a_n = a_n</math></center> | ||
Revision as of 14:16, 24 July 2006
Vieta's formulas are a set of equations relating the roots and the coefficients of polynomials.
Introduction
Let 
be a polynomial of degree 
, so 
,
where the coefficient of 
is 
and 
. 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
The coefficient of 
in this expression will be the 
th symmetric sum of the 
.
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 
.
