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let h be the set of polynomial functions divisible by (x-2) . proof that h is a subspace of vector space of polynomials. show that the three conditions for a subspace are satisficed.

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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:

  1. 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.
  2. 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.
  3. 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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