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I am not fully understanding how using substitutions helps in finding equations with roots related to a given equation. For example, in the following problem (extract from CIE A-Level Further Math paper):

The equation x^4 - x^3 -1 =0 has roots (alpha, beta, gamma, delta). By using the substitution y=x^3, find the exact value of (alpha^6+beta^6+gamma^6+delta^6)...

Please help...

I'm doing Further Maths in A-Level and I can't find a single resource about centroids of areas and volumes using integration. In one of the examiners' reports to a past paper I was solving I saw a reference to some formulae. I searched high and low to no avail. Please do help by either giving me a link to a site/video/anything that explains in detail the concept of centroids (for both areas and volumes) and the use of integration for it, or by explaining it yourself.

Eagerly awaiting a response.

DG = DH - TDS.

Here, DS is the entropy of the system, and not that including the surrounding, right? So, how can an exothermic reaction cause an increase in entropy of the system? Shouldn't the fact that energy is being given out by the system result in a decrease of thermal and positional entropy of the system? Could someone please help make this clear...?

I wanted to know whether the integral of the cross-sectional area gives the amount of the material used or the amount of air the solid of revolution can contain (i.e. hollow volume). I am not sure, but I feel it measures the amount of material used, in which case, what do I do to find the inside volume? Subtract the material volume from the whole volume (if you get what I mean)???

I've already started learning japanese and am getting along pretty well with it. but i also want to learn chinese, so i was wondering which would be better to learn first.