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MHT CET202523 Apr 2025Morning ShiftMathematicsQuadratic EquationActual

If x=-2+ -3 , then the value of 2 x^4+5 x^3+7 x^2-x+38 is equal to

Options

  1. A1
  2. B-2
  3. C3
  4. D5

Correct answer

C. 3

Step-by-step solution

Given x = -2 + -3 = -2 + i 3 . To find the minimal polynomial, observe that x + 2 = i 3 . Squaring both sides yields (x + 2)^2 = (i 3 )^2 , which simplifies to x^2 + 4x + 4 = -3 , giving the quadratic equation x^2 + 4x + 7 = 0 . The polynomial P(x) = 2x^4 + 5x^3 + 7x^2 - x + 38 is divided by x^2 + 4x + 7 : 2x^4 + 5x^3 + 7x^2 - x + 38 = (x^2 + 4x + 7)(2x^2 - 3x + 5) + 3 Since x^2 + 4x + 7 = 0 , substitution gives P(x) = 0 (2x^2 - 3x + 5) + 3 = 3 . Final answer: 3

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