Question

Which of the following subsets in P₃ is a subspace of P₃= a₀ + a₁x + a₂x² + a₃x³ : A. The set of all polynomials with degree less than or equal to 3.

B. The set of all polynomials with exactly 3 terms.

C. The set of all polynomials with a leading coefficient of 1.

D. The set of all polynomials with only even powers of x.

Answer 1

The subset of all polynomials with degree less than or equal to 3 is a subspace of P₃, but the subsets of polynomials with exactly 3 terms, a leading coefficient of 1, and only even powers of x are not subspaces of P₃.

Step-by-step explanation:

To determine which of the given subsets is a subspace of P₃, we need to check if each subset satisfies the three conditions of a subspace:

1. Additivity: If p and q are polynomials in the subset, then p + q must also be in the subset.
2. Scalar Multiplication: If p is a polynomial in the subset and c is a scalar, then c * p must also be in the subset.
3. Containment: The subset must contain the zero polynomial.

Let's evaluate each subset:

A. The set of all polynomials with degree less than or equal to 3.

This subset satisfies all three conditions. The sum of two polynomials with degree less than or equal to 3 is still a polynomial with degree less than or equal to 3. Similarly, multiplying a polynomial with degree less than or equal to 3 by a scalar will result in a polynomial with degree less than or equal to 3. Finally, the zero polynomial is included in this subset.

B. The set of all polynomials with exactly 3 terms.

This subset does not satisfy the additivity condition because the sum of two polynomials with exactly 3 terms may have a different number of terms. Therefore, this subset is not a subspace of P₃.

C. The set of all polynomials with a leading coefficient of 1.

This subset does not satisfy the scalar multiplication condition because multiplying a polynomial with a leading coefficient of 1 by a scalar will result in a polynomial with a different leading coefficient. Therefore, this subset is not a subspace of P₃.

D. The set of all polynomials with only even powers of x.

This subset satisfies all three conditions. The sum of two polynomials with only even powers of x will still have only even powers of x. Multiplying a polynomial with only even powers of x by a scalar will not change the powers of x. The zero polynomial, which consists of only even powers of x, is also included in this subset.