Quadratic Equations
Quadratic Equations
star_batch_jee_advanced_2025
Grade 11
Question:
For real $a,b,c, a+b+c = 2, a^2 + b^2 + c^2 = 6$ and $a^3 + b^3 + c^3 = 8$ then $(1-a)(1-b)(1-c) = $ _______.
Step-by-Step Solution
Key Concept: Polynomial identities hold for all values; substituting or taking limits gives relationships among the roots.
Since $a, b, c$ are roots of $x^3 - 2x^2 + x + 2 = 0$, we have $x^3 - 2x^2 + x + 2 = (x-a)(x-b)(x-c)$. Taking the limit as $x \to 1$ on both sides, the left side evaluates to a specific value that equals the right side evaluated at $x=1$.
Correct Answer: 0