Final answer:
The space of third-order polynomials is a vector space where vector addition is commutative, meaning that the order of addition does not affect the final sum. This property can be demonstrated by choosing any three vectors A, B, and C, finding their sum A + B + C, and then adding them in a different order to show the result is the same.
Step-by-step explanation:
The space of third-order polynomials, with addition defined in the usual way for polynomials, is a vector space. Vector addition is commutative, meaning that the order in which vectors are added does not affect the final sum. To show this property, we can choose any three vectors A, B, and C, all with different lengths and directions, and find the sum A + B + C. Then, we can add them in a different order and show that the result is the same. For example, let A = 2x^3 + 3x^2 + 4x, B = -x^3 - 2x^2 + x, and C = 5x^3 - x. The sum A + B + C is (2 - 1 + 5)x^3 + (3 - 2 + 0)x^2 + (4 + 1 - 1)x = 6x^3 + 1x^2 + 4x. Now, let's add them in a different order, B + C + A, which gives (-1 + 5 + 2)x^3 + (-2 + 0 + 3)x^2 + (1 - 1 + 4)x = 6x^3 + 1x^2 + 4x. As we can see, the result is the same, demonstrating the commutative property of vector addition.