2007 UNCO Math Contest II Problems/Problem 8

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Problem

A regular decagon $P_1P_2P_3\cdots P_{10}$ is drawn in the coordinate plane with $P_1$ at $(2,0)$ and $P_6$ at $(8,0)$. If $P_n$ denotes the point $(x_n ,y_n )$, compute the numerical value of the following product of complex numbers: $( x_1+iy_1)( x_2+iy_2)( x_3+iy_3) \cdots (x_{10} + iy_{10})$ where $i^2 = -1$ as usual.

[asy] draw(polygon(10),dot); draw((-2,0)--(2,0),black); draw((-5/3,-2)--(-5/3,2),black); MP("P_1",(-1,0),NW);MP("(2,0)",(-.9,0),SW); MP("P_6",(1,0),NE);MP("(8,0)",(.9,0),SE); [/asy]


Solution

$9,706,576$

Translate the center of the decagon to the origin. Now the vertices represent the roots of $f(x)=x^{10}-3^{10}=0$. Since the $P_n$ are each $5$ more than the roots of $f(x) = 0$ , they would be the roots of$f(x-5)=0$ or $(x-5)^{10}-3^{10}=0$. The product then is the constant term, or $5^{10}-3^{10}= 9,706,576$

See Also

2007 UNCO Math Contest II (ProblemsAnswer KeyResources)
Preceded by
Problem 7
Followed by
Problem 9
1 2 3 4 5 6 7 8 9 10
All UNCO Math Contest Problems and Solutions
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