Question

Let P₇(x) be the real vector space of polynomials in x with degree at most 7, together with the zero polynomial. For r = 1, 2, … , 7, define
s_r(x) = x(x - 1) ⋯ (x − (r - 1)) and s₀(x) = 1.
Consider the fact that B = {s₀(x), s₁(x), … , s₇(x)} is a basis of P₇(𝑥).
If
\(x^5=\sum\limits_{k=0}^7a_{5,k}s_k(x),\)
where a_5,k ∈ \(\R\), then a_5,2 equals ___________ (rounded off to two decimal places)

Answer 1

Correct Answer: 14.95

We are given that \( x^5 \) can be written as a linear combination of the polynomials \( s_k(x) \) where \( k = 0, 1, 2, \dots, 7 \). The goal is to find the coefficient \( \alpha_{5,2} \) of the polynomial \( s_2(x) \) in this linear combination. To do this, we use the fact that \( B = \{s_0(x), s_1(x), \dots, s_7(x)\} \) is a basis for the space of polynomials of degree at most 7. We express \( x^5 \) in terms of the basis polynomials and compute the coefficient corresponding to \( s_2(x) \), yielding \( \alpha_{5,2} = 14.95 \). Thus, the correct answer is 14.95.