Thread: Non-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.

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

I just wish to know if the following reasoning is correct?

Working few steps as given by I-Think above:

Originally Posted by I-Think

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)

Now, since x=-1 is a root of LHS but not of RHS, LHS cannot be equal to RHS for any values of d,e and f. Is it correct to conclude like this?

Yes, it is a very correct and much shorter and neater way of answering the question, for case 1.

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.