Suppose that $ S = \{ p_1, p_2, p_3, p_4 \} $, where:

\[ p_1(x) = 220 + 21x + 45x^2 + 161x^3 + 21x^4 + 293x^5 \]

\[ p_2(x) = -156 - 28x + 293x^2 + 250x^3 - 164x^4 - 208x^5 \]

\[ p_3(x) = -102 + 299x + 263x^2 - 164x^3 - 37x^4 - 69x^5 \]

\[ p_4(x) = 150 + 21x - 47x^2 + 17x^3 - 77x^4 - 157x^5 \]

and

\[ p(x) = -354 + 23740x + 37454x^2 + 9619x^3 - 8864x^4 + 5512x^5 \]

The polynomial $ p $ is a linear combination of $ S $ written in the form $ \alpha p_1 + \beta p_2 + \gamma p_3 + \delta p_4 $.

Find a possible set of values for $ \alpha, \beta, \gamma, \delta $.

To find the values of α, β, γ, and δ, we need to solve the equation αp1 + βp2 + γp3 + δp4 = p. This equation can be solved by setting up a system of equations using the coefficients of each polynomial.

Explanation:

The polynomial p can be written as a linear combination of the polynomials in S by using the variables α, β, γ, and δ. To find the values of α, β, γ, and δ, we need to solve the equation αp1 + βp2 + γp3 + δp4 = p. This equation can be solved by setting up a system of equations using the coefficients of each polynomial. By equating the coefficients of each term on both sides of the equation, we can find the values of α, β, γ, and δ.

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