2024 · Q88
If x3 + px2 + qx + r is an integer for all integral values of x, then consider the following statements :
- Ip must be an integer
- IIq must be an integer
- IIIr must be an integer Which of the statements given above is/are correct?
Elementary MathematicsAlgebra
- AI and II only
- BIII only
- CI, II and III
- DNone of the statements is correct
Explanation
x = 0 forces r ∈ ℤ, so III must hold. But p = q = 1/2 gives x³+(x²+x)/2+r, and x(x+1)/2 is always integral, so the polynomial is integer-valued for all integral x with non-integral p, q — statements I and II need not hold. Only III is correct — option B.