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Tuesday, April 3, 2018

Problem of the Week 4-3-18: Polynomial Root Product Minimum [Polynomials]

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Solution below.

Solution


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1 comment:

  1. f(i)f(-1) = (i-x1)(i-x2)(i-x3)(i-x4)(-1-x1)(-i-x2)(-i-x3)(-i-x4)
    = (1+x1^2)(1+x2^2)(1+x3^2)(1+x4^2)

    It also equals:
    = (1 - ai - b + ci + d)(1 + ai - b - ci + d)
    = (1 - (b-d))^2 + (a-c)^2
    = ((b-d) - 1)^2 + (a-c)^2

    You can minimize over u=b-d, subject to u >= 5 and v=a-c. They're both upward facing parabolas, so the minimum will be the closest possible point to it's center. This gives u=5 and v=0.

    = (5 - 1)^2 + 0^2 = 16.

    Btw, one trivial solution would be x1=x2=x3=x4=1 or:

    f(x) = x^4 + 4x^3 + 6x^2 + 4x + 1

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