Multiple choice

Equation $x^4+ax^3+bx^2+cx+1=0$ has real roots ($a,b,c $ are non-negative). Minimum non-negative real value of $b$ is

  1. $12$
  2. $15$
  3. $6$
  4. $10$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For x^4 + ax^3 + bx^2 + cx + 1 = 0, by AM-GM, the roots must be 1 (if all are positive). If roots are 1, 1, 1, 1, then (x-1)^4 = x^4 - 4x^3 + 6x^2 - 4x + 1 = 0. Here b=6.

AI explanation

For a polynomial with all real roots to minimize b while having a non-negative real value, we assume the roots occur in reciprocal pairs (t, 1/t) to keep the constant term 1. Expanding (x-t)^2(x-1/t)^2 gives a coefficient b = t^2 + 1/t^2 + 4. We rewrite b as (t - 1/t)^2 + 6, which reaches its absolute minimum when the squared term is zero. Thus, the minimum non-negative real value of b is 6.

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