Show that there do not exist polynomials p(x) and q(x) of degree greater than or equal to 1 with integer co-efficients such that
p(x)q(x)=x^5+ 2x+1.
If , then either
Case 1
is of degree and is of degree ,
or
Case 2
is of degree and is of degree
I'll do case 1
But the polynomials have integer coefficients, so and
(I'm going to choose for both to simplify things here)
Expanding
Comparing coefficients
, so
, so
, so
, so
Contradiction has occurred, so Case 1 is impossible
Case 2 is proven similarly
Maybe another solution(only highlights):
x^5+ 2x+1=0 have only one real root (wolframalpha or calculus laws).
Suppose that m is root of x^5+ 2x+1, now, we need to prove that m is not rational.
Lets say that m=p/q , where p and q are integers.
m^5+ 2m+1=(p/q)^5+ 2(p/q)+1=0
==>
p^5+2p*q^4+q^5=0
Integer solution only when p=q=0
Contradiction! Therefore, m is irrational.