this is a question on polynomials but has A.P. involved in it:

by zeroes, we mean the roots of the polynomial where f(x) = 0

let a,b,c be the roots of the polynomial

then a+b+c = p.........(1)

ab+bc+ca = q............(2)

and abc = r................(3)

if a,b,c are in A.P. then we know that 2b = a+c

putting in (1)  => b= p/3

from (2)   b(a+c) + ca = q =>  2b² + ac =q => ac = q- (2/9)p²

putting the values of b and ac in (3)

( q- (2/9)p²) . p/3 = r