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 COMPLETE IN ADDITION
--ITS ASSOCIATIVE
--IT HAS IDENTITY ELEMENT ZERO
--FOR EVERY POLYNOMIAL THERE IS AN ADDITIVE INVERSE WHICH WILL ALSO HAVE EVEN DEGREE
--ALSO ADDITION IS ABLIAN
--ITS COMPLETE IN MULTIPLICATION
--ITS ASSOCIATIVEI IN MULTIPLICATION
GR9768 Question # 57
Make it easy! 
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.
Those are all correct.--ITS ASSOCIATIVE
--IT HAS IDENTITY ELEMENT ZERO
--FOR EVERY POLYNOMIAL THERE IS AN ADDITIVE INVERSE WHICH WILL ALSO HAVE EVEN DEGREE
--ALSO ADDITION IS ABLIAN
--ITS COMPLETE IN MULTIPLICATION
--ITS ASSOCIATIVE IN MULTIPLICATION
That is wrong!! It is not closed under addition:--ITS COMPLETE IN 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.