2011/092) When is one of the roots of a cubic equal to the product of the other two roots?

Find, in terms of the coefficients, a necessary and sufficient condition for one of the roots of ax³ + bx² + cx + d = 0 (a ≠ 0) to be equal to the product of the other two roots.

Let the roots be p,q,pq

So a(x-p)(x-q)(x-pq)

= a x^3 – ax^2(p + q + pq) + ax(pq + pq^2 + p^2q) – ap^q^2

Comparing coefficients

b/a = -(p+q+pq) .. 1

c/a = pq + pq^2+ p^2q = pq(1+p+q) … 2

d/a = -p^2 q^2 ..3

subtract 1 from both sides of (1)

b/a - 1 = - (p+1)(q+1)

from (2 ) and (3)

c/a - d/a = pq(1+p+q+pq) = -pq(b/a- 1)

or c-d = - pq(b-a)

square both sides

(c-d)^2 = p^2q^2(b-a)^2 = -d(b-a)^2

Or (c-d)^2 + d(b-a)^2= 0