# Are the polynomials of form $a_0 + a_1x + a_2x^2 +a_3x^3$, with $a_i$ rational, a subspace of $P_3$?

I am trying to prove whether a specific form of polynomials form a subspace of $P_3$. I know that any set $W$ forms a subspace of a vector space $V$ if the set $W$ is closed under addition and scalar multiplication (is this right? please clarify).

Based on this, are polynomials of form $a_0 + a_1x + a_2x^2 + a_3x^3$ where $a_0,a_1,a_2,a_3$ are rationals, a subspace of $P_3$ ? I am not sure how any polynomials of this form actually violates the closure properties.

• "closed under addition and scalar multiplication" AND nonempty. Oct 16, 2017 at 4:10
• What is "P3"? Is it the space of all real polynomials of degree $\leq 3$? Oct 16, 2017 at 4:11
• Do you really mean $a_2x^2a_3x^3$, as you have written? or do you mean $a_2x^2+a_3x^3$? Oct 16, 2017 at 4:12
• @Hayden Yes. Polynomials of degree <= 3 Oct 16, 2017 at 4:13
• @GerryMyerson I have edited the post to reflect a2x^2 + a3x^3 Oct 16, 2017 at 4:14

• @AbrarHossain Close. They are in $P_3$ but not in your proposed subspace. ..
Hint: Are all scalar multiples of $1$ ($= 1+0x+0x^2+0x^3$) in the set $$\{ a_0+a_1x+a_2x^2+a_3x^3 \mid a_0,a_1,a_2,a_3 \in \mathbb{Q}\}$$ ?
• e.g. $\pi \cdot 1 = \pi$ Oct 16, 2017 at 4:34