Multiple choice

If the equation $\displaystyle x^{4} - 4x^{3} + ax^{2} + bx + 1 = 0$ has four roots (all positive), then value of $\displaystyle a$ and $\displaystyle b$, are given as

  1. $\displaystyle a = 4, \: b = 6$
  2. $\displaystyle a = -4, \: b = 6$
  3. $\displaystyle a = 6, \: b = -4$

  4. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For x^4 - 4x^3 + ax^2 + bx + 1 = 0, let roots be r1, r2, r3, r4. Product = 1, Sum = 4. If all roots are 1, then (x-1)^4 = x^4 - 4x^3 + 6x^2 - 4x + 1 = 0. Thus a=6, b=-4.

AI explanation

By Vieta's formulas for the equation x^4 - 4x^3 + ax^2 + bx + 1 = 0, the sum of the four positive roots is 4 and the product of the roots is 1. Applying the AM-GM inequality to these four positive numbers, their arithmetic mean (which is 1) must be greater than or equal to their geometric mean (which is also 1), meaning equality must hold. This restricts all four roots to be exactly 1. Substituting the roots into the symmetric sums gives a = 6 and b = -4.

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