Which of the following are subspaces of P3

a) U={f(x) | f(x) in P3, f(2)=1}
b) U={xg(x) | g(x)i in P2}
c) U={x(g(x) | g(x) in P3}
d) U={xg(x) + (1-x)h(x) | g(x) and h(x) in P2}
e) U= The set of all polynomials in P3 with constant term 0
f) U= U={f(x) | f(x) in P3, degf(x)=3}

a) No; not closed under addition.
If f, g are in U, then (f+g) is in U, because (f+g)(2) = f(2) + g(2) = 1 + 1 ≠ 1.
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b) Yes; check closure under addition and scalar multiplication.
Let x f(x), x g(x) be in U (with f, g in P2).
Then, x f(x) + x g(x) = x(f(x) + g(x)), and f+g being in P2 ==> x (f(x) + g(x)) is in U.

If k is a scalar, then k * (x f(x)) = x * (k f(x)) and k f(x) is in P2 ==> k * (x f(x)) is in U.
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c) Yes; see part b.
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d) Yes, see part b.
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e) Yes, adding two such polynomials or multiplying such a polynomial by a scalar does not introduce a nonzero constant term.
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f) No, not closed under addition:
x^3 and -x^3 + x are in U, but x^3 + (-x^3 + x) = x is not in U.

I hope this helps!