Let P 7 (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 s 0 (x) = 1. Consider the fact that B = {s 0 (x), s 1 (x), … , s 7 (x)} is a basis of P 7 (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)

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)

Updated On: Aug 13, 2024

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The correct answer is 14.95 to 15.05. (approx)

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