Re: quadratics with related roots



Adam Atkinson wrote:
>
> [...]
>
> If the roots of the quadratic equation (insert quadratic here)
> are alpha and beta, write down a quadratic whose roots are (two
> formulas involving alpha and beta), without solving for alpha
> and beta.
>
> I've seen cases like asking for roots alpha^2 and beta^2
> or alpha/beta and beta/alpha.

Usually those kinds of problems are applications of the fact
that any symmetric polynomial of n variables can be expressed
as a polynomial in solely the elementary symmetric polynomials
of those variables.

Elementary symmetric (ES) polynomials of a set of n variables
{a_i} are the coefficients of powers of x (including x^0, i.e.
the constant term) in the expansion of:

(x - a_1).(x - a_2)... (x - a_n)

So for a quadratic with roots a_1, a_2 its 2 ES polynomials are
a_1 + a_2 and a_1.a_2, which we can denote by B_1 and B_0 resp.

If you're looking for a quadratic with roots a_1^2 and a_2^2
it's coefficients up to sign are a_1^2 + a_2^2 and a_1^2.a_2^2,
denoted by say C_1 and C_0 resp, which are symmetric polynomials
in a_1, a_2 and thus, by the result mentioned in the first para
above, both polynomials in B_1 and B_0. Sure enough its easy to
see that C_1 = B_1^2 - 2.B_0 and C_0 = B_0^2

.

Relevant Pages

  • Re: quadratics with related roots
    ... >> formulas involving alpha and beta), ... >> and beta. ... >> I did maths at university and don't recall ever having do to anything ... manipulate roots of polynomials without ever knowing the value of them. ...
    (sci.math)
  • Re: Sphere packing in six dimensions, root system for E6, Wikipedia article
    ... in Phi_E8 with alpha and beta orthogonal, ... counting orthogonal roots as usual. ... "Essays in the history of Lie groups and algebraic groups", ...
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  • Re: quadratics with related roots
    ... > If the roots of the quadratic equation are ... > formulas involving alpha and beta), ... > I did maths at university and don't recall ever having do to anything ...
    (sci.math)
  • Re: quadratics with related roots
    ... are alpha and beta, write down a quadratic whose roots are, without solving for alpha and beta. ... or alpha/beta and beta/alpha. ... that any symmetric polynomial of n variables can be expressed as a polynomial in solely the elementary symmetric polynomials of those variables. ...
    (sci.math)
  • Re: quadratics with related roots
    ... > If the roots of the quadratic equation are ... > formulas involving alpha and beta), ... > I did maths at university and don't recall ever having do to anything ...
    (sci.math)