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Cubic root derivation help

Hello, thanks for entering my thread :biggrin: All help is appreciated :smile:

I have recently discovered a result which I was not aware of.
We have three roots: a,b,c
Where m=-(a+b+c) and n=ab + bc + ac
& x^3 +mx^2 + nx + p=0

I have tested that this result is true, but I am struggling to find a proof or derivation on the internet. If possible could someone please explain or send me a relevant link. Also does anyone have any hints/tips to remember this solution? It seems very useful but also very easily forgotten.


Thanks :biggrin: :biggrin:
There is a more general form of this result in FP1:
If px^3+qx^2+rx+s=0:
a+b+c = -q/p
ab+bc+ca = r/p
abc = -s/p
(You can verify that if p=1 then the first two formulae reduce to your results).

These are called Vieta's formulas. Proof from Wikipedia:
Vieta's formulas can be proved by expanding the equality:












(which is true since are all the roots of this polynomial), multiplying the factors on the right-hand side, and identifying the coefficients of each power of

Formally, if one expands the terms are precisely where is either 0 or 1, accordingly as whether is included in the product or not, and k is the number of that are excluded, so the total number of factors in the product is n (counting with multiplicity k) – as there are n binary choices (include or x), there are terms – geometrically, these can be understood as the vertices of a hypercube. Grouping these terms by degree yields the elementary symmetric polynomials in – for xk, all distinct k-fold products of

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