Algebra
Polynomial Equations, Transformation of Roots
jee_adv_2026_mock_p1
Grade 12
Question:
If the roots of the equation x^3 - 3x^2 + 3x + 7 = 0 are α, β, γ, find the value of (α-1)(β-1)(γ-1).
Step-by-Step Solution
Key Concept: Transform the equation by substituting y = x - 1.
Step 1: Let y = x - 1 => x = y + 1. Step 2: Substitute: (y+1)^3 - 3(y+1)^2 + 3(y+1) + 7 = y^3 + 8. Step 3: So y^3 + 8 = 0 => y^3 = -8. Step 4: Product of roots of y^3 + 8 = 0 is -8 (since for y^3 + 0y^2 + 0y + 8 = 0, product = -8). Thus (α-1)(β-1)(γ-1) = -8.
Correct Answer: -8
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