If the sum of the zeroes of the quadratic polynomial kx^2 + 3x + k = 0 is equal to their product, then k = -1.
To see this, let the two zeroes of the polynomial be x1 and x2. Then the polynomial can be written as k(x - x1)(x - x2) = 0. Since the product of the zeroes is equal to their sum, we have x1x2 = x1 + x2. Rearranging this equation gives x1 + x2 - x1x2 = 0.
Comparing this equation to the given polynomial, we see that the coefficient of x2 on the left side (1) is equal to the coefficient of x2 on the right side (k), and the constant term on the left side (-x1x2) is equal to the constant term on the right side (k). Therefore, k = 1 and k = -x1x2.
Since the coefficient of x2 must be the same on both sides, k = -1.