Math Contest #23 Results and Solution

Solution

The problem of this contest was to solve a polynomial:
x⁴ + 8x³ + 24x² + 32x + 15 = 0

This might look pretty hard, but if you know what to look for you might notice a regularity there.
First of all you need to add 1:
x⁴ + 8x³ + 24x² + 32x + 16 = 1
And now the magic happens when you remove the factors typical for to the binomial theorem:
x⁴ + 4*2x³ + 6*4x² + 4*8x + 16 = 1
x⁴ + 4*2¹x³ + 6*2²x² + 4*2³x + 2⁴ = 1
According to the binomial theorem that is equal to:
(x+2)⁴ = 1
taking the 4th root leads to the four solutions:
x₁+2 = 1 → x₁ = -1,
x₂+2 = -1 → x₂ = -3,
x₃+2 = i → x₃ = i-2,
x₄+2 = -i → x₄ = -i-2

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List of participants with their entries:

Name	solution found	comment
@crokkon	correct
@zuerich	correct

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Winner draw:

Not necessary since the reward can be divided be the number of entries:
Congratulations @zuerich and @crokkon , you won 1 SBI each!
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The next contest starts in 2 days. Don't miss it!