To prove that the set of polynomial functions divisible by (x-2) is a subspace of the vector space of polynomials h, we need to show that it satisfies the three conditions for a subspace:
- The set must contain the zero vector. In this case, the zero vector is the polynomial function 0, which is clearly divisible by (x-2). Therefore, this condition is satisfied.
- The set must be closed under addition. This means that if f(x) and g(x) are both elements of the set (i.e. they are both polynomial functions divisible by (x-2)), then their sum f(x) + g(x) must also be an element of the set. Since the sum of two polynomial functions divisible by (x-2) is also divisible by (x-2), this condition is satisfied.
- The set must be closed under scalar multiplication. This means that if f(x) is an element of the set and k is a scalar, then the product kf(x) must also be an element of the set. Since the product of a polynomial function is divisible by (x-2) and a scalar is also divisible by (x-2), this condition is satisfied.
Therefore, since the set of polynomial functions divisible by (x-2) satisfies all three conditions for a subspace, h is indeed a subspace of the vector space of polynomials.
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