Final answer:
The set of all polynomials such that p(0) = 0 is a subspace because it contains the null vector (the zero polynomial), is closed under addition, and is closed under scalar multiplication.
Step-by-step explanation:
To determine if the given set is a subspace of the space of polynomials, we must check three conditions. The set must contain the null vector, be closed under addition, and be closed under scalar multiplication. In this case, the null vector for the space of polynomials is the zero polynomial, where all coefficients are zero, which indeed has the property p(0) = 0. This satisfies the first condition.
For the second condition, if we take any two polynomials in the set, say p(x) and q(x), both satisfying p(0) = 0 and q(0) = 0, their sum (p+q)(x) will also satisfy (p+q)(0) = p(0) + q(0) = 0 + 0 = 0. Hence, the set is closed under addition.
Lastly, for scalar multiplication, if c is any scalar and p(x) is any polynomial in the set, then (cp)(0) = c*p(0) = c*0 = 0, which means the set is closed under scalar multiplication. Therefore, the set of all polynomials such that p(0) = 0 is indeed a subspace of the space of polynomials.