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tmua.me Extreme Set A Paper 2 · Q10

Paper 2Algebra

A polynomial P(x)P(x) with real coefficients has degree 1111, and P(1)≠0P(1)\ne0. Every real root is simple: whenever P(r)=0P(r)=0, the expression (x−r)2(x-r)^2 is not a factor of P(x)P(x).

It is also given that, for every real root rr of P(x)P(x),

P ⁣(11−r)=0.P\!\left(\frac1{1-r}\right)=0.

Which is the complete list of possible numbers of distinct real roots of P(x)P(x)?

A33 only
B33 and 66 only
C66 and 99 only
D33 and 99 only
E33, 66 and 99 only
F11, 33, 55, 77, 99 and 1111

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