Suppose that u and v are real numbers for which u + iv has modulus sqrt7. Express the real part of (u+iv)^-4 in terms of a polynomial in v.
First, expand (u + iv)^4 (we can take the reciprocal at the end):
(u + iv)^4 = (u^2 + 2iuv - v^2)(u^2 + 2iuv - v^2)
= u^4 + 2ivu^3 - u^2v^2 + 2ivu^3 - 4u^2v^2 - 2iuv^3 -u^2v^2 - 2iuv^3 + v^4
Ignore the terms containing i, since we're told we only need to deal with the real part:
u^4 - 6u^2v^2 + v^4
Remember that |u + iv| = sqrt(7), so u^2 + v^2 = 7. Rearrange that to give u^2 = 7-v^2 and substitute in:
= (7-v^2)^2 - 6(7-v^2)v^2 + v^4
= 49 - 14v^2 + v^4 - 42v^2 + 6v^4 + v^4
= 49 - 56v^2 + 8v^4
So the real part of (u+iv)^-4 is equal to:
(49 - 56v^2 + 8v^4)^-1 in terms of v.
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