Q. The equation `(x-3)^9 + (x-3^2)^9 + ...................+(x-3^9)^9 = 0` has
A) all roots real
B) one real and rest imaginary roots
C) atleast one real root
D) none of these
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The equation given in the problem is: (x - 3)^9 + (x - 3^2)^9 +...(x - 3^9)^9 = 0
If x - 3 is negative, each of the other terms is also negative and their sum cannot be equal to 0. If x - 3^9 is positive, the other terms are also positive and again the sum is not equal to 0.
For one particular value of x lying between 3 and 3^9, some of the terms are positive and the rest negative. As all of them are raised to an odd power the sign of the terms remains the same. This allows their sum to be equal to 0. There is only one such real value. The equation can have other roots that are imaginary.
The correct answer is option B, one real root and the rest imaginary.
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