Q60 / 160 · AP EAMCET 2020 (17 Sep Shift 2) · Mathematics

Equation \(x^5-5 x^3+5 x^2-1=0\) has equal roots

  • A
    2
  • Correct answer
    B
    3
  • C
    4
  • D
    5

Solution

Given, \(f(x)=x^5-5 x^3-5 x-1=0\) Now by inspection, \(x=1 \text { is a zero of } f(x)\) So, we can divide \(f(x)\) by \(x-1\) to get, \(f(x)=(x-1)\left(x^4+x^3-4 x^2+x+1\right)\) We similarly proceed to get \(\begin{aligned} & f(x)=(x-1)(x-1)\left(x^3+2 x^2-2 x-1\right) \\ & f(x)=(x-1)(x-1)(x-1)\left(x^2+3 x+1\right) \end{aligned}\) So, \(f(x)\) has 3 equal roots.

Hence, the correct answer is option: 'B'.

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