Final answer:
The first set of polynomials cannot be linearly independent because it exceeds the number of basis elements for P₂, while the second set has the correct number and can be tested for linear independence and spanning properties by analyzing linear combinations and the span of the vectors.
Step-by-step explanation:
To address whether the sets of polynomials {2, x², x, 2x+3} and {p₁=1+x, p₂ =x²−x, p₃=1+2x²} are linearly independent and span P₂, we need to use the concepts of spanning sets and linear independence in the context of vector spaces. For a set of vectors (or polynomials) to span a vector space, any vector (or polynomial) within that space should be expressible as a linear combination of those vectors (or polynomials). For linear independence, no vector (or polynomial) in the set can be written as a linear combination of the others.
For the first set, since the highest degree is two and there are four polynomials, we can argue without calculation that the set cannot be linearly independent because there should only be three polynomials in a basis for P₂, representing the coefficients of x², x, and the constant term. Thus, the set does not both span P₂ and are linearly independent. For the second set, we have three polynomials, which is the correct number for a basis of P₂. To determine if they are linearly independent, we could set up a linear combination equal to zero and solve for the coefficients. If the only solution is trivial (all coefficients are zero), then the polynomials are linearly independent. If they are also non-zero multiples of x², x, and the constant term, they span P₂.