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Finding Quadratic (Posted on 2019-04-12) Difficulty: 3 of 5
Find a quadratic polynomial which is a factor of x101+x94+x57+x33-1

No Solution Yet Submitted by Danish Ahmed Khan    
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Some Thoughts Another approach Comment 2 of 2 |
Let f(x) = x^101 + x^94 + x^57 + x^33 - 1 and let g(x) = (x-1) * f(x) = x^102 - x^101 + x^95 - x^94 + x^58 - x^57 + x^34 - x^33 - x + 1

Start with the assumption that x^2+x+1 is a factor of f(x).  If that is true then x^3-1 will be a factor of g(x).

Rearrange g(x) into (x^102-x^57) - (x^101-x^95) - (x^94-x^58) + (x^34-x) - (x^33-1).  Each of the five binomials in this arrangement has x^3-1 as a factor, therefore g(x) has x^3-1 as a factor, which then implies that x^2+x+1 is a factor of f(x).

  Posted by Brian Smith on 2019-04-15 11:59:51
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