the index of x^4 in each vector needs to be -2000

Given the additional constraint that the coefficient of \(x^4\) in each polynomial basis vector must be -2000, we need to adjust our basis vectors accordingly. Let's construct the basis:

We start by ensuring that each polynomial \(p(x)\) in the basis satisfies \( p(1) = p'(1) = 0 \) and the coefficient of \(x^4\) is -2000.

Consider the following basis vectors:

1. \( p_1(x) = -2000(x-1)^2 x \)
2. \( p_2(x) = -2000(x-1)^3 x \)
3. \( p_3(x) = -2000(x-1)^2 x^2 \)

These polynomials satisfy the conditions \( p(1) = p'(1) = 0 \), and the coefficient of \(x^4\) is -2000 in each case.

Therefore, the basis \( B \) is:

\[ B = \{ -2000(x-1)^2 x, -2000(x-1)^3 x, -2000(x-1)^2 x^2 \} \]

Explore how to adjust polynomial basis vectors ensuring the coefficient of x^4 is precisely -2000. This problem involves constructing polynomials satisfying specific conditions at x = 1, useful for advanced undergraduate study.

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