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Question

For the polynomial equation
(x+q)(x+r)(q−p)(r−p)+(x+r)(x+p)(r−q)(p−q)+(x+p)(x+q)(p−r)(q−r)−1=0, where p,q,r are distinct real numbers.
Which of the following statement is correct?

A
It has 2 distinct real roots.
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B
It has no real roots.
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C
It has real and equal roots.
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D
It has more than 2 real roots.
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Solution

The correct option is D It has more than 2 real roots.
(x+q)(x+r)(qp)(rp)+(x+r)(x+p)(rq)(pq)+(x+p)(x+q)(pr)(qr)1=0
Putting x=q
(q+r)(q+p)(rq)(pq)1=011=0
Similarly,
x=r,p are also the roots of the polynomial.
Therefore, the given equation is an identity, so it has more than 2 roots.

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