Polynomials question (1 Viewer)

[pikachu975]

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?

 

[jathu123]

Well you could form a polynomial with roots alpha^2, beta^2, gamma^2, delta^2. And then finding the sum of roots (-b/a) of that polymomial

 

[InteGrand]

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?

 

[jathu123]

Ah thanks for that, didn't know this works for more than three terms

 

[pikachu975]

Is there a formula for the sum of cubes? Thanks for this

 

[Shadowdude]

Is there a formula for the sum of cubes? Thanks for this

yes

i derived it from scratch during my honours year... a fun exercise

 

[InteGrand]

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)).

 

[InteGrand]

You may want to keep in mind the three variable result

 

[fluffy54324]

Is there any way to do all this stuff using pythagorus? I mean with all the a^2, b^2

 

[Shadowdude]

Is there any way to do all this stuff using pythagorus? I mean with all the a^2, b^2

uhhh good pattern recognition, but pythagoras is for triangles/orthogonality only

 

[fluffy54324]

Why thankyou my good sir. I tip my hat to you