Difference between revisions of "Vieta's Formulas"
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<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> | <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> | ||
| − | The coefficient of <math> x^k </math> in this expression will be the <math>k </math>th [[elementary symmetric sum]] of the <math>r_i</math>. | + | The coefficient of <math> x^k </math> in this expression will be the <math>k</math>-th [[elementary symmetric sum]] of the <math>r_i</math>. |
We now have two different expressions for <math>P(x)</math>. These must be equal. However, the only way for two polynomials to be equal for all values of <math>x</math> is for each of their corresponding coefficients to be equal. So, starting with the coefficient of <math> x^n </math>, we see that | We now have two different expressions for <math>P(x)</math>. These must be equal. However, the only way for two polynomials to be equal for all values of <math>x</math> is for each of their corresponding coefficients to be equal. So, starting with the coefficient of <math> x^n </math>, we see that | ||
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More commonly, these are written with the roots on one side and the <math>a_i</math> on the other (this can be arrived at by dividing both sides of all the equations by <math>a_n</math>). | More commonly, these are written with the roots on one side and the <math>a_i</math> on the other (this can be arrived at by dividing both sides of all the equations by <math>a_n</math>). | ||
| − | If we denote <math>\sigma_k</math> as the <math>k</math>th elementary symmetric sum, then we can write those formulas more compactly as <math>\sigma_k = (-1)^k\cdot \frac{a_{n-k}}{a_n{}}</math>, for <math>1\le k\le {n}</math>. | + | If we denote <math>\sigma_k</math> as the <math>k</math>-th elementary symmetric sum, then we can write those formulas more compactly as <math>\sigma_k = (-1)^k\cdot \frac{a_{n-k}}{a_n{}}</math>, for <math>1\le k\le {n}</math>. |
== See Also == | == See Also == | ||
Revision as of 22:12, 2 July 2016
Vieta's Formulas, otherwise called Viète's Laws, are a set of equations relating the roots and the coefficients of polynomials.
Introduction
Vieta's Formulas were discovered by the French mathematician François Viète.
Vieta's Formulas can be used to relate the sum and product of the roots of a polynomial to its coefficients. The simplest application of this is with quadratics. If we have a quadratic 
with solutions 
and 
, then we know that we can factor it as
(Note that the first term is 
, not 
.) Using the distributive property to expand the right side we get
We know that two polynomials are equal if and only if their coefficients are equal, so means that 
and 
. In other words, the product of the roots is equal to the constant term, and the sum of the roots is the opposite of the coefficient of the 
term.
A similar set of relations for cubics can be found by expanding 
.
We can state Vieta's formula's more rigorously and generally. 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 elementary 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 elementary symmetric sum, then we can write those formulas more compactly as 
, for 
.
