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) + (-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.