Let r₁, r₂, r₃ be the roots of the cubic equation x^3 - x^2 + 2x - 1 = 0 . If S_k = r₁^k + r₂^k + r₃^k for any integer k 1 , then the quadratic equation whose roots are S₂ and S₄ is :

Step-by-step solution

For the cubic equation x^3 - x^2 + 2x - 1 = 0 , we have: Sum of roots: S₁ = r₁ + r₂ + r₃ = 1 Sum of product of roots taken two at a time: r₁ r₂ = 2 Product of roots: r₁ r₂ r₃ = 1 First, we find S₂ = r₁^2 + r₂^2 + r₃^2 : S₂ = (r₁ + r₂ + r₃)^2 - 2(r₁ r₂ + r₂ r₃ + r₃ r₁) S₂ = (1)^2 - 2(2) = 1 - 4 = -3 To find S₄ , we can use the characteristic equation. Since r is a root, it satisfies: r^3 - r^2 + 2r - 1 = 0 r^3 = r^2 - 2r + 1 Multiplying by r , we get: r^4 = r^3 - 2r^2 + r Substituting r^3 = r^2 - 2r + 1 into the abo