## GR9768 Question # 57

Forum for the GRE subject test in mathematics.
mobius70
Posts: 19
Joined: Thu Jan 24, 2008 1:11 am

### GR9768 Question # 57

Let R be field of real numbers and R[x] the ring of polynomials in x with coefficients in R. Which of the following subsets of R[x] is a subring of R?

I) All polynomials whose coefficient of x is zero.
II) All polynomials whose degree is even integer, together with zero polynomial.
III) All polynomials whose coefficient are rational numbers.

THE ANSWER SAYS ONLY I AND III ARE CORRECT.
I DON'T GET WHY II IS NOT CORRECT.

--ITS ASSOCIATIVE
--IT HAS IDENTITY ELEMENT ZERO
--FOR EVERY POLYNOMIAL THERE IS AN ADDITIVE INVERSE WHICH WILL ALSO HAVE EVEN DEGREE
--ITS COMPLETE IN MULTIPLICATION
--ITS ASSOCIATIVEI IN MULTIPLICATION

lime
Posts: 129
Joined: Tue Dec 04, 2007 2:11 am
Make it easy!
--ITS ASSOCIATIVE
--IT HAS IDENTITY ELEMENT ZERO
--FOR EVERY POLYNOMIAL THERE IS AN ADDITIVE INVERSE WHICH WILL ALSO HAVE EVEN DEGREE
--ITS COMPLETE IN MULTIPLICATION
--ITS ASSOCIATIVE IN MULTIPLICATION
Those are all correct.
That is wrong!! It is not closed under addition:

Two polynomials x^2 and -x^2+x are both of even degree = 2. But their sum
(x^2) + (-x^2 + x) = x
is of degree 1.
Last edited by lime on Wed Mar 12, 2008 3:44 pm, edited 2 times in total.

mobius70
Posts: 19
Joined: Thu Jan 24, 2008 1:11 am
Thx lime ..
You seem to be good here .. :->
just out of curosity ..are you going for some GRE or soemthing