Question

Let P₁₂(x) be the real vector space of polynomials in the variable x with real coefficients and having degree at most 12, together with the zero polynomial. Define
\(𝑉 = \left\{𝑓 ∈ 𝑃_{12}(𝑥): 𝑓(−𝑥) = 𝑓(𝑥) \text{for all}\ 𝑥 ∈ ℝ\ \text{and}\ 𝑓(2024) = 0\right\}.\)
Then, the dimension of V is _____________

Answer 1

Correct Answer: 6

The space \( V \) consists of even polynomials (polynomials for which \( f(-x) = f(x) \)) of degree at most 12, and the condition \( f(2024) = 0 \) further restricts the possibilities. The even polynomials of degree at most 12 form a 6-dimensional subspace of \( P_{12}(x) \), as they are spanned by \( 1, x^2, x^4, x^6, x^8, x^{10} \). Thus, the dimension of \( V \) is 6.