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JEE MainMathematicsQuadratic Equation

Let , , be the roots of the equation x^3 - 4x^2 + x - 6 = 0 . If P_n = ^n + ^n + ^n for all natural numbers n , then the value of P₈ - 4P₇ + P₆ P₅ is equal to

Options

  1. A-6
  2. B4
  3. C6
  4. D1

Correct answer

C. 6

Step-by-step solution

Since , , are the roots of the equation x^3 - 4x^2 + x - 6 = 0 , they must satisfy it. For the root , we have: ^3 - 4 ^2 + - 6 = 0 Multiplying the entire equation by ^5 , we obtain: ^8 - 4 ^7 + ^6 - 6 ^5 = 0 Similarly, for the roots and , we get: ^8 - 4 ^7 + ^6 - 6 ^5 = 0 ^8 - 4 ^7 + ^6 - 6 ^5 = 0 Adding these three equations vertically yields: ( ^8 + ^8 + ^8) - 4( ^7 + ^7 + ^7) + ( ^6 + ^6 + ^6) - 6( ^5 + ^5 + ^5) = 0 Using the given definition P_n = ^n + ^n + ^n , this can be written as: P₈ - 4P₇ + P₆ - 6P₅ = 0 R

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