A Bagel with Root Probability and a Kappa coffee

Algebra Level 3

Let κ ( x ) = x 3 + a 2 x 2 + z j x + a 0 \kappa(x)=x^3+a_2x^2+z^jx+a_0 for natural number j j , nonzero integers a 2 a_2 and a 0 a_0 , and that rational number z z satisfies the equation z j = a 0 a 2 z^j=\frac{a_0}{a_2} . What is the probability that κ \kappa has irrational roots?

9/5/2019: correction made, thanks to comment below by: Joe Mansley United Kingdom, 18 years old About: I'm weird.

2 3 \frac{2}{3} 1 4 \frac{1}{4} 1 2 \frac{1}{2} 1 3 \frac{1}{3}

Frank Giordano by Frank Giordano , 57, USA

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1 solution

Frank Giordano
Mar 7, 2018

G-filtered Polycules

0 Helpful 0 Interesting 0 Brilliant 0 Confused

You should state in the problem that z is rational since that is necessary for the result to hold.

Joe Mansley - 1 year, 9 months ago

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good point, radical two to the natural exponent of 2 is a ratio of integers 12 and 6. thanks!

Frank Giordano - 1 year, 9 months ago

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