Schaum's probably means that some quadratics are irreducible over the reals. Of course, as you've pointed out, all quadratics are reducible over the complexes.
The fraction can be broken down into + with A and B to be determined.
You can solve this using algebra as it's normally taught. However since the denominator consists of linear factors, then there's a formula that says A = 1/(2-1) = 1 and B = 1/(1 - 2) = -1. More generally if you have , then A = (b - a) and B = (a - b).
Your first puzzle is given an expression , break it up into the individual fractions and determine what A, B and C are without first using algebra (you might see a pattern in connection with the previous example).
Your second puzzle is given the expression , again break it up into the individual fractions and determine what A, B and C are in this case (please note I take exception with Schaum's that says some quadratics are irreducible since they all
are reducible as the last problem illustrates).
The point to this exercise is that it's far simpler to do these problems by formula than by algebra (also note that this particular method works with fractions with one as the numerator and only linear factors in the denominator).
Okay sharpen your pencils and go to work.
I guess it's time for me to solve the puzzles but first a correction. The general formula for a two-termed linear denominator is A = 1/(b - a) and B = 1/(a - b).
Regarding the puzzle, the general formula in both cases, is the same which is A = 1/(b - a)(c - a), B = 1/(a - b)(c - b), C = 1/(a - c)(b - c) so letting a = 1, b = 2 and c = 5, then the answer is A = 1/(2 - 1)(5 - 1) = 1/8, B = 1/(1 - 2)(5 - 2) = -1/3 and C = 1/(1 - 5)(2 - 5) = 1/12.
Here's another pattern . . .
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
Let's check the fourth equation.
On the right, get a common denominator:
Do not muiltiply it out!
Don't you recognize the numerator?
. . It is: . . . . . . Ha!
I think it's covered in most books (or rather at least those which I've seen) and if it's omitted it's probably
because it's judged to be rather limiting -- the cases where it works are only those which could be done
without any particular methods anyway; it doesn't work in where the method of partial fractions matters!