A Rational Cubic with Imaginary Entry Point

Algebra Level pending

Consider the cubic integer polynomial h ( x ) = 3 x 3 + a 2 x 2 + a 1 x + a 0 h(x)=3x^3+a_2x^2+a_1x+a_0 given prime ∣ a 0 ∣ > 15 |a_0|>15 and where h ( 1 ) = 12 h(1)=12 . Find a 0 a_0 such that h ( x ) h(x) has one rational root and two complex roots.


The answer is 17.

Frank Giordano by Frank Giordano , 57, USA

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1 solution

Frank Giordano
Oct 25, 2016

G-filtered Polycules

0 Helpful 0 Interesting 0 Brilliant 0 Confused

what's interesting is that the first available prime, given the condition prime ∣ a 0 ∣ > 15 |a_0|>15 , is the answer. We'll see if this is the case for the similar quartic polynomial.

Frank Giordano - 4 years, 7 months ago

get the latest version of "G-filtered Polycules" here: https://www.facebook.com/groups/factorthis/

Frank Giordano - 4 years, 7 months ago

https://brilliant.org/problems/prime-quartic-with-k-narrative/?ref_id=1268042

interesting that the cubic finite sum polynomial has possible rational roots minus one and plus or minus one third. will the quartic possible rational roots yield some kind of pattern ???

Frank Giordano - 4 years, 5 months ago

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