If x^3+px^2+qx+r is an integer for all integral values of x, then consider the following statements :<br>I. p must be an integer<br>II. q must be an integer<br>III. r must be an integer<br>Which of the statements given above is/are correct?

Explanation

Let P(x) = x^3+px^2+qx+r. For x=0, P(0)=r, which must be an integer (Statement III is true). For x=1 and x=-1, p+q \in \mathbb{Z} and p-q \in \mathbb{Z}, which means 2p \in \mathbb{Z}. So p and q can be half-integers. For example, if p=\frac{1}{2} and q=\frac{1}{2}, P(x) = x^3 + \frac{x^2+x}{2} + r = x^3 + \frac{x(x+1)}{2} + r, which is always an integer since x(x+1) is always even. Thus, p and q do not have to be integers.

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