Question

Let W be the subset of P₃ , consisting of all polynomials p(x) with p(−2)=0. (a) What are the three criteria for W to be a subspace of P₃ ?

Answer 1

W is a subspace of P₃ if it includes the zero polynomial, is closed under vector addition, and is closed under scalar multiplication. These criteria ensure that the subset W has the necessary properties to be considered a subspace.

Step-by-step explanation:

To determine if W is a subspace of P₃, it must satisfy three criteria:

1. The zero polynomial must be in W. This is because every subspace must include the zero vector (or zero element), which in the case of P₃ is the zero polynomial, denoted typically by 0(x) = 0. Since 0(−2) = 0, the zero polynomial indeed belongs to W, satisfying this criterion.
2. W must be closed under vector addition, which in this context means that if two polynomials p(x) and q(x) are in W, then their sum p(x) + q(x) must also be in W. Since both p(−2) and q(−2) are zero, it follows that (p + q)(−2) = p(−2) + q(−2) = 0 + 0 = 0, so the sum is also in W.
3. W must be closed under scalar multiplication, meaning that if p(x) is in W and c is any scalar, then cp(x) must also be in W. Since p(−2) = 0, it follows that (cp)(−2) = c*p(−2) = c*0 = 0, so cp(x) is in W as well.