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.

Printable View

- Nov 16th 2011, 01:48 AMmalaygoelNon-existence
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. - Nov 17th 2011, 08:17 AMI-ThinkRe: Non-existence
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 - Nov 17th 2011, 01:01 PMmalaygoelRe: Non-existence
- Nov 18th 2011, 09:04 AMI-ThinkRe: Non-existence
Yes, it is a very correct and much shorter and neater way of answering the question, for case 1.

- Nov 18th 2011, 11:08 AMAlso sprach ZarathustraRe: Non-existence
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.