Let P 2 be the vector space of all polynomials of degree less than or equal to 2, under the usual vector addition and scalar multiplication. Let S = {f (x) ∈ P 2 | if deg(f ) = 2, then f has two complex roots}.  (a) Is "0 ∈ S. Why or why not? (b) Is S closed under addition? If S is closed under addition, prove it. If S is not closed under addition, provide a counterexample.   (c) Is S closed under scalar multiplication? If S is closed under scalar multiplication, prove it. If S is not closed under scalar multiplication, provide a counterexample.

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