Question

Determine whether the polynomials x² + 1, x² - 1, x² + x + 1 span P(2).

Answer 1

The polynomials x² + 1, x² - 1, x² + x + 1 do not span P(2).

Step-by-step explanation:

To determine whether the polynomials x² + 1, x² - 1, x² + x + 1 span P(2), we need to check whether every polynomial in P(2) can be expressed as a linear combination of these three polynomials.

Let's start by checking if x² + 1 can be expressed as a linear combination of x² - 1 and x² + x + 1.

If we subtract x² - 1 from x² + 1, we get 2. This means that 2 cannot be expressed as a linear combination of x² - 1 and x² + x + 1.

Therefore, the polynomials x² + 1, x² - 1, x² + x + 1 do not span P(2).