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TSR Wiki > Study Help > Subjects and Revision > Mathematics > STEP > STEP III 1997 question 3 solution

The roots of the polynomial are and hence the polynomial can be written.

.

Dividing by , and converting the formula for the sum of a geometric sequence into and actual sum of a geometric sequence, we get

as required.

Next part: Embedding the problem in the Argand plane, we can let be represented by the complex numbers , respectively. the point O then corresponds to the complex number 0.

But then, let in the above proved formula. Then, the LHS becomes zero, and the sum in the RHS is precisely the sum of the position vectors of , which must therefore also be zero, as required.

Let the point have coordinates , the point have coordinates .

Then the coordinates of for are given by .

Now we have

But

(This is basically the horizontal bit of the vector identity proved in the last part)

So

as required.

Solution by ukgea