What is the dimension of the vector space $\mathbb Q [x]$/$((x+1)^2)$

Question

What is the dimension of the ring $\mathbb Q [x]$$/$$((x+1)^2)$ as a vector space over $\mathbb Q$ .

How to deal with this types of problems.

Answer 1

Hint: The polynomial ring $\mathbb{Q}[x]$ is an infinite-dimensional vector space with basis $x^i$ for $i \geq 0$. Now you are considering a certain quotient, which means that you identify some things. Which degrees of polynomials "remain" in your case? This will allow you to find a basis and hence to solve your problem

Answer 2

Hint:Use the fact that $\Bbb {Q}[x]$ is an Euclidean domain and polynomial can be expressed in division algorithm form.Count the form of remainder which we can get when divisor is the polynomial $(x+1)^2$.