Polynomials question (1 Viewer)

pikachu975

Premium Member
Joined
May 31, 2015
Messages
2,739
Location
NSW
Gender
Male
HSC
2017
For roots of a polynomial stuff, do you have to remember like alpha^2 + beta^2 + gamma^2 + delta^2 and what this is equivalent to? I know alpha^2 + beta^2 = (alpha+beta)^2 - 2alpha*beta but do we have to remember the hard ones like shown above? If we do is there a way to derive them?
 

InteGrand

Well-Known Member
Joined
Dec 11, 2014
Messages
6,109
Gender
Male
HSC
N/A
For roots of a polynomial stuff, do you have to remember like alpha^2 + beta^2 + gamma^2 + delta^2 and what this is equivalent to? I know alpha^2 + beta^2 = (alpha+beta)^2 - 2alpha*beta but do we have to remember the hard ones like shown above? If we do is there a way to derive them?




 

InteGrand

Well-Known Member
Joined
Dec 11, 2014
Messages
6,109
Gender
Male
HSC
N/A
Is there a formula for the sum of cubes? Thanks for this
Yes, there is. In fact, there is a recursive formula for all sums of non-negative integer powers (called \textbf{power sums}) in terms of lower powers' sums and elementary symmetric polynomials (expressions like sum of roots, sum of pairs of roots, etc., which are easily computable given a polynomial P(x)). But you're not expected to know them for the HSC. If you want to read up about them, you can do so here:

https://en.wikipedia.org/wiki/Newton's_identities .

As background reading too, you may want to see:

- https://en.wikipedia.org/wiki/Symmetric_polynomial

- https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial (and this part: https://en.wikipedia.org/wiki/Eleme..._fundamental_theorem_of_symmetric_polynomials (which was asked as a question by glittergal96 in a previous MX2 Advanced Level marathon here: http://community.boredofstudies.org...-marathon-advanced-archive-7.html#post7068974)).
 

Users Who Are Viewing This Thread (Users: 0, Guests: 1)

Top