The set of all polynomials of degree exactly n fails one of

The set of all polynomials of degree exactly n fails one of the closure prop-
erties of a vector space. Which one, and why?

Solution

Solution :- The set of all polynomials of degree exactly n fails one of the closure properties of a vector space.

Let us see the example below.

f = x⁵ + x 1 and g = x⁵ are in V but f + g = (x ⁵ + x 1) x⁵ = x 1 is not in V.