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 $\displaystyle p(x)q(x)=x^5+ 2x+1$, then either
Case 1
$\displaystyle p$ is of degree $\displaystyle 1$ and $\displaystyle q$ is of degree $\displaystyle 4$,
or
Case 2
$\displaystyle p$ is of degree $\displaystyle 2$ and $\displaystyle q$ is of degree $\displaystyle 3$
I'll do case 1
$\displaystyle (ax+b)(cx^4+dx^3+ex^2+fx+g)=x^5+2x+1$
But the polynomials have integer coefficients, so $\displaystyle a=c=\pm{1}$ and $\displaystyle b=g=\pm{1}$
(I'm going to choose $\displaystyle +1$ for both to simplify things here)
$\displaystyle (x+1)(x^4+dx^3+ex^2+fx+1)=x^5+2x+1 $
Expanding
$\displaystyle x^5+(d+1)x^4+(e+d)x^3+(e+f)x^2+(f+1)x+1=x^5+2x+1$
Comparing coefficients
$\displaystyle d+1=0$, so $\displaystyle d=-1$
$\displaystyle e+d=0$, so $\displaystyle e=1$
$\displaystyle e+f=0$, so $\displaystyle f=-1$
$\displaystyle f+1=2$, so $\displaystyle f=1$
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.