Haven't taken a formal math class in a long time. Obviously roots of polynomial equations is a basic Algebra topic, but if you scroll down to like pages 3-4, I'm pretty sure I never saw anything like that in an Algebra class:

http://assets.cambridge.org/97811084...75_excerpt.pdf

Is it just a more rigorous approach to Algebra? What would a class that has this stuff be called in US curriculum?

Equations have solutions, expressions have roots, and it annoys me when these terms are conflated. Am I wrong about this?

not if you can answer my op

I'm sure you have, at least if you have ever factorized quadratics. For Example 1.1, the quadratic x^2 - 2px + p factorizes as (x - a)(x - 3a), and then you have to solve a simultaneous equation to find p.

Yeah I do remember that, but I don't recall ever using sigma notation during that part of math. Like the stuff they're doing below that, they have have what looks like expanding a squared sum (except replacing the "b" term with the beta symbol) which I remember. But then they equate that to some formula using sigma notation, I don't remember anything like that.

Or like below in example 1.2, I remember doing binomial expansion. But on the third line of that example ("sum for each possible root"), I don't remember using sigma notation in those kind of problems. Then again I know the binomial theorem uses sigma notation so maybe I just forgot it since our teacher just told us to use the pascal's triangle thing. Frankly I'm not even sure what they mean by "sum for each possible root".

Or like when they talk about a "recurrence relation", is that the same thing I learned as a recursive formula? The way they use it there seems foreign (no pun intended) to me.

Or just the fact that there's specific sections on how to deal with cubics and quartics, I know I never did anything like that in algebra. I mean obviously there were 3rd and 4th degree polynomials but we never approached them as specific areas of study. I remember there being formulas for like cubed sums/differences/etc. but nothing generalized for all cubics/quartics.

So I'm trying to figure out how much of this is due to difference in notation/language, how much is due to differences in what's actually learned in US/UK curriculum, or how much is due to whatever else.

Neither am I. School math textbooks in the UK are not well written.

Probably. I just googled the difference and it seems unclear in general.

You're more right than wrong.

A solution is a value of the variable or variables that makes the equation true.

I'm not sure that I would say an "expression" has a root, but instead say that "functions" have roots. This is a little nit-picky, but there exist functions for which we don't have an expression (a combination of well-defined mathematical symbols).

This notation is awful and non-standard.

"The product can be written as $\sum \alpha \beta = \alpha \beta$."

That looks like a complete mess to me. They switched to their non-standard sigma notation and then called it something that it almost certainly cannot be.

Just to help out a little here, their notation is a bit wonky. But when they talk about "sum of possible roots," etc. They're referring to things like Vieta's formulas.

For example, take the equation

$$x^7 + 9x - 3.$$

I don't care WHAT the roots are, but I can tell you that their product is $3.$ How can I do that? Well, we know this must factor as

(x - blah)(x-blah)(x-blah)(x-blah)(x-blah)(x-blah)(x-blah)

Where "blah" are the roots. When you multiply that all out, the constant term is just the product of all the (-blah)'s. That's means you can find the product of the roots by looking at the constant term.

Similarly, you can find the sum of the roots from one of the terms (I won't tell you which, think about it -- you want only one root at a time, none multiplied together, so that might help -- or just look at some examples).

That doesn't square up at all with their language and notation in example 1.2. In that problem, the instruction reads "Find $\alpha^3 + \beta^3$ in summation notation." There's nothing to do with the sum of roots of a polynomial function.