Calculating the Intersection of Subspaces in Vector Spaces

jimmycricket
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Given two subspaces of the vector space of all polynomials of at most degree 3 what is the general method to calculate the intersection of the two subspaces?
 
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What is the general method to calculate the intersection of two subspaces of any vector space? It's not even clear what you mean when you say the word "calculate" - are you looking for a basis? If you have a specific problem or example in mind it would be a lot easier if you tell us what it is.
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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