Algebra|MCR3UO|

3. Suppose that f(x) = x3 􀀀 px2 + qx and g(x) = 3x2 􀀀 2px + q for some positive integers p and q. (a) If p = 33 and q = 216, show that the equation f(x) = 0 has three distinct integer solutions and the equation g(x) = 0 has two distinct integer solutions. (b) Suppose that the equation f(x) = 0 has three distinct integer solutions and the equation g(x) = 0 has two distinct integer solutions. Prove that (i) p must be a multiple of 3, (ii) q must be a multiple of 9, (iii) p2 􀀀 3q must be a positive perfect square, and (iv) p2 􀀀 4q must be a positive perfect square. (c) Prove that there are innitely many pairs of positive integers (p; q) for which the following three statements are all true:  The equation f(x) = 0 has three distinct integer solutions.  The equation g(x) = 0 has two distinct integer solutions.  The greatest common divisor of p and q is 3 (that is, gcd(p; q) = 3).