Question

Let W ⊆ P be the subset of polynomials that have 3 as a root. (So a polynomial is in W if,

when we plug 3 in for x, we get zero.) Prove that W is a subspace of P using the definition of a
subspace.

Answer 1

W is a subspace of P because it is non-empty (contains the zero polynomial), it is closed under addition (the sum of any two polynomials in W also has 3 as a root), and it is closed under scalar multiplication (multiplying any polynomial in W by a scalar also has 3 as a root).

Step-by-step explanation:

To show that W is a subspace of P, the space of polynomials, we must verify that (1) W is non-empty, (2) W is closed under addition, and (3) W is closed under scalar multiplication.

Firstly, W contains the zero polynomial, which has 3 as a root, hence W is non-empty. Secondly, if p(x) and q(x) are in W, then p(3) = 0 and q(3) = 0. For any two polynomials in W, their sum (p + q)(x) also satisfies (p + q)(3) = p(3) + q(3) = 0, proving closure under addition. Lastly, for any scalar a and any polynomial p(x) in W, the product (ap)(x) results in (ap)(3) = a * p(3) = a * 0 = 0, establishing closure under scalar multiplication. Therefore, W is a subspace of P as it satisfies all three conditions of the subspace test.