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Is $ax^3 + bx^2 +cx + d$, for which $a$, $b$, $c$, $d$ are integers, subset of P3. I got that it was a subspace but the book got that it wasn't a subspace and now I'm confused.

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I take it that you want $bx^2$ instead of bx^3 and P3 is probably the space of polynomials of degree at most $3$ with, say, real coefficients. If so, Hint: is the subset closed under taking scalar multiples? – t.b. Mar 15 '11 at 4:36
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What field is your vector space over? Is the set of all such polynomials closed under scalar multiplication?

The answer to this question will show why the set of these polynomials do not form a subspace.

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It's not a subspace because when you multiply by a scalar (I assume $P_3$ is the vector space of polynomials of degree at most 3, with real coefficients), you get polynomials that are not necessarily in that subset, i.e., polynomials whose coefficients are not integers. For instance, $\sqrt{2}(x^3+x^2+x+1)=\sqrt{2}x^3+\sqrt{2}x^2+\sqrt{2}x+\sqrt{2}$, the coefficients of this polynomial are no longer integers.

When checking for subspaces, make sure you don't forget multplication by scalars (you only checked that the subset was closed under vector addition).

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