Question

Show that for each natural number n
(x-1)(x-2)(x-3)......(x-n)-1 is irreducible over the rationals
Please, help

Answer 1

@eliassaab

Answer 2

Eh,... @rational solves rational problem??? hehehe, give me your hand, please

Answer 3

lol you want to prove there are no rational roots for that polynomial right

Answer 4

yes, I think induction works well, right?

Answer 5

Clearly \(1,2,3,\ldots, n\) are not roots of the given polynomial as each of them leave a remainder \(-1\).

Answer 6

yes

Answer 7

for \(x\gt n\), the polynomial evaluates to some integer \(\gt 0\) proving there are no integer solutions

Answer 8

Are we allowed to use rational root thm ?

Answer 9

I think I got it.
let P(x) = (x-1)(x-2).....(x-n) -1
suppose P(x) has a rational root p/q, then by rational root test, we must have q| a_n which is the first leading coefficient of P(x) , but the product of monic polynomial is a monic one also, that gives us \(a_n =1\)
hence q|1 iff \(q=\pm 1\) that turns root to integer
Hence, P(x) has no rational root.
Am I right?

Answer 10

-1 at the end of P(x) is a constant, it changes the constant term of the previous product only, right?
I mean if P(x) =(x-1)(x-2) -1 = x^2-3x+1 , the "monic" property doesn't change because of it , -1.

Answer 11

Looks good! q = +- 1 narrows the search to integers

Answer 12

thank you:)

Answer 13

hey you need to say something about negative roots also as we have just considered x >=1