Problem 34 Divisibility and primes challenge set.

[mcallist]

Hi Ron,
Can you explain how the polynomial gets simplified here please?

(n^4 - 5n^2 +4)

In this question it goes from that to (n^2 -4)(n^2-1).
I don't follow how it gets there.

Will you break it down please?

Thanks,
Thomas

[RonPurewal]

1/
First, verify that the result is actually correct. I.e., multiply the expression (n^2 - 4)(n^2 - 1) back out, and check that you actually get the original expression. (This should always be step #1 in apprehending any unfamiliar-looking thing: Make sure it actually works!)

2/
I'm going to assume that you understand why
x^2 - 5x + 4 = (x - 1)(x - 4).
(If you don't, then you've got some groundwork to do in factoring basic polynomials, before you start thinking about the sort of animal that's in this problem.)

If you understand that, then you should understand everything else in this progression:
x^2 - 5x + 4 = (x - 1)(x - 4).
Q^2 - 5Q + 4 = (Q - 1)(Q - 4).
(pink flamingo)^2 - 5(pink flamingo) + 4 = (pink flamingo - 1)(pink flamingo - 4).
whatever^2 - 5(whatever) + 4 = (whatever - 1)(whatever - 4).

This problem is just the same thing, with "whatever" = x^2. Just realize that the square of x^2 is x^4, and it should make sense.