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AP EAMCET202422 May 2024Morning ShiftMathematicsQuadratic EquationActual

If and are two distinct negative roots of x^5-5 x^3+5 x^2-1=0 , then the equation of least degree with integer coefficients having - and - as its roots is

Options

  1. Ax^2-3 x+1=0
  2. B-x^4+5 x^2-5 x+1=0
  3. C-x^4-5 x^2+5 x+1=0
  4. Dx^4-3 x^2+1=0

Correct answer

D. x^4-3 x^2+1=0

Step-by-step solution

x^5-5 x^3+5 x^2-1=0 By inspection x=1 is a root aligned & (x-1) (x^4+x^3-4 x^2+x+1 )=0 & (x-1)^2 (x^3+2 x^2-2 x-1 ) 0 & (x-1)^3 (x^2+3 x+1 )=0 & x=1, x= -3 5 2 & = -3+ 5 2 , = -3- 5 2 aligned Since imaginary roots occurs in pair. So, equation having roots -i , i , i ,-i aligned & (x-i )(x+i )(x-i )(x+i )=0 & x^4+( + ) x^2+ =0 x^4-3 x^2+1=0 aligned

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