Multiple choice

The roots of the equation ${x}^{3}-{9x}^{2}+23\ x-15=0$, if integers, are in

  1. $A.P$.
  2. $G.P$.
  3. $A.G.P$.

  4. None of these
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A Correct answer
Explanation

Testing integer roots for x^3 - 9x^2 + 23x - 15 = 0: x=1 gives 1-9+23-15=0. Dividing by (x-1) gives x^2 - 8x + 15 = 0, which factors to (x-3)(x-5). Roots are 1, 3, 5. These are in A.P. with common difference 2.

AI explanation

Using Vieta's formulas for the cubic equation x^3 - 9x^2 + 23x - 15 = 0, the sum of the roots alpha + beta + gamma is 9 and the product of the roots alpha * beta * gamma is 15. Testing integer factors of 15 that sum to 9 gives the roots 1, 3, and 5. Since the differences between consecutive terms are constant (3 - 1 = 2 and 5 - 3 = 2), the roots are in arithmetic progression.

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