I have to use Gauss's lemma to show that
if $P(x) = a_nx^n+....+a_1x+a_0$ in $\mathbb Z[x]$ has a rational root r/s, where gcd (r,s) =1, then $s|a_n$ and $r|a_0$
My problem is: I can prove it without using Gauss's lemma.
How to link my proof to Gauss's lemma? Please, help
My proof is in comment box

Question

I have to use Gauss's lemma to show that 
if $P(x) = a_nx^n+....+a_1x+a_0$ in $\mathbb Z[x]$ has a rational root r/s, where gcd (r,s) =1, then $s|a_n$ and $r|a_0$
My problem is: I can prove it without using Gauss's lemma.
How to link my proof to Gauss's lemma? Please, help
My proof is in comment box

loser66 · Answer

Suppose r/s is a rational root of P(x) , then P(r/s)=0
$a_n(r/s)^n+......+a_1(r/s)+a_0=0\a_0= -(a_n(r/s)^n+.....+a_1r/s\a_0=r/s(-a_n(r/s)^{n-1}-....-a_1)$
the second term is a polynomial but $a_0$ is a number, hence only 1 possibility is r/s | a_0

loser66 · Answer

$r/s| a_0$ --> $r = a_0 s$ that is $r|a_0$

rational · Answer

Basically you want to prove rational root theorem using Gauss lemma

rational · Answer

whats Gauss's lemma anyways ?

loser66 · Answer

Gauss's lemma: Let R be a UFD with field of fractions F
a) The product of 2 primitive elements of R[x] is primitive
b) Suppose f(x) in R[x] .Then f(x) has a factorization f(x) = g(x) h(x) in F[x] with degree of g, h greater or equal 1 if and only if f(x) has such a factorization in R[x]

loser66 · Answer

We can't use part a of lemma here, since we don't know whether P(x) is primitive or not.

rational · Answer

I found this http://en.wikipedia.org/wiki/Rational_root_theorem they're dividing gcd of the coefficients of polynomial to make it primitive

rational · Answer

click this http://en.wikipedia.org/wiki/Rational_root_theorem#Proof_using_Gauss.27s_lemma

loser66 · Answer

Thank you so much. It is waaaay better than my proof. :)