Cubic Dilemma

Algebra Level 4

There is a smallest positive real number a a such that there exists a positive number b b such that the roots of the polynomial x 3 − a x 2 + b x − a x^3-ax^2+bx-a are all real. In fact, for this value of a a the value of b b is unique. Find the value of b b .


The answer is 9.

William Isoroku by William Isoroku , 25, USA

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1 solution

Otto Bretscher
Feb 11, 2016

With the given information, " b b is unique", we know that the polynomial must have a triple root x = c x=c . Thus x 3 − a x 2 + b x − a = ( x − c ) 3 = x 3 − 3 c x 2 + 3 c 2 x − c 3 x^3-ax^2+bx-a=(x-c)^3=x^3-3cx^2+3c^2x-c^3 so c = a 3 , a 2 = 27 c=\frac{a}{3}, a^2=27 and b = 3 c 2 = 9 b=3c^2=\boxed{9} .

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