Q13/006)Let f(x) = (x-a) (x-b) (x+a) (x+b). Find all integers a and b such that f(x) = a^2 b^2 has exactly 3 distinct integer roots.

As it is a degree 4 polynomial it has 4 root so it must have a double toot and 2 single roots

f(x) = (x-a) (x-b) (x+a) (x+b) = a^2b^2
=> (x^2 - a^2)(x^2 - b^2) = a^2b^2
=> x^4 - (a^2 + b^2)x^2 + a^2b^2 = a^2b^2
=> x^2 [x^2 - (a^2 + b^2)] = 0
=> x = 0 or x = ±√(a^2 + b^2)

This has a double root 0 and 2 more roots when a,b are sides of a Pythagorean triangle or a or b ( but not both) = 0