Minimum Value of Polynomial Coefficient

CAT 2008 Slot 1 · QA · Medium · Algebra

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If the roots of the equation x3−ax2+bx−c=0x^3 - ax^2 + bx - c = 0 are three consecutive integers, then what is the smallest possible value of bb?

  1. A.

    −13-\frac{1}{\sqrt{3}}

  2. B.

    −1-1

  3. C.

    00

  4. D.

    11

  5. E.

    13\frac{1}{\sqrt{3}}

Answer

B

Explanation

Let the roots of the cubic equation be α−1,α,α+1\alpha - 1, \alpha, \alpha + 1.

The coefficient bb represents the sum of product of roots taken two at a time: b=(α−1)α+α(α+1)+(α−1)(α+1)b = (\alpha - 1)\alpha + \alpha(\alpha + 1) + (\alpha - 1)(\alpha + 1) b=α2−α+α2+α+α2−1=3α2−1b = \alpha^2 - \alpha + \alpha^2 + \alpha + \alpha^2 - 1 = 3\alpha^2 - 1

Since α\alpha is an integer, the minimum value of 3α2−13\alpha^2 - 1 occurs when α=0\alpha = 0, giving b=−1b = -1.

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