Final answer:
The subsets of P2 that are subspaces are: A. The set of all constant polynomials, C. The set of all quadratic polynomials, and D. The set of all cubic polynomials.
Step-by-step explanation:
To determine which subsets of P2 are subspaces, we need to consider the three requirements for a set to be a subspace:
- It must contain the zero vector.
- It must be closed under addition.
- It must be closed under scalar multiplication.
A. The set of all constant polynomials is a subspace of P2 because it meets all three requirements. The zero vector is the constant polynomial 0, and any constant polynomial added to another constant polynomial will result in another constant polynomial. Similarly, multiplying a constant polynomial by a scalar will yield another constant polynomial.
B. The set of all linear polynomials is not a subspace of P2 because it does not contain the zero vector. The zero vector is the polynomial 0x + 0, and the set of all linear polynomials does not include this polynomial.
C. The set of all quadratic polynomials is a subspace of P2 because it meets all three requirements. The zero vector is the polynomial 0x^2 + 0x + 0, and any quadratic polynomial added to another quadratic polynomial will result in another quadratic polynomial. Similarly, multiplying a quadratic polynomial by a scalar will yield another quadratic polynomial.
D. The set of all cubic polynomials is also a subspace of P2 because it meets all three requirements. The zero vector is the polynomial 0x^3 + 0x^2 + 0x + 0, and any cubic polynomial added to another cubic polynomial will result in another cubic polynomial. Similarly, multiplying a cubic polynomial by a scalar will yield another cubic polynomial.