Consider the quotient spaces obtained by reducing the space P of polynomials modulo various subspaces. If M = P_n is P/M finite dimensional? What if M is the subspace consisting of all even polynomials? What if M is the subspace consisting of all polynomials divisible by $x_n $( where$ x_n(t) = t^n)$
Please, I appreciate any help. :)

Question

Consider the quotient spaces obtained by reducing the space P of polynomials modulo various subspaces. If M = P_n is P/M finite dimensional? What if M is the subspace consisting of all even polynomials? What if M is the subspace consisting of all polynomials divisible by $x_n $( where$ x_n(t) = t^n)$
Please, I appreciate any help. :)

ybarrap · Answer

In the 1st two cases I can see how you can put the sets on a 1-1 correspondence with the Natural numbers, which means they are NOT finite. For example, in the second case map 2n to  n for n=1,2,.., which puts even numbers in a 1-1 correspondence with the natural numbers. 

For polynomials divisible by t^n I'm not sure.