Find the zeroes of the quadratic polynomial:
To find the zeroes of the quadratic polynomial f(x) = abx^2 + (b^2 – ac)x – bc, we need to solve the equation f(x) = 0. The zeroes of a polynomial are the values of x for which the polynomial equals zero.

Using the quadratic formula:
The quadratic formula states that for a quadratic equation of the form ax^2 + bx + c = 0, the zeroes can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / 2a

Applying the quadratic formula to the given polynomial:
For the polynomial f(x) = abx^2 + (b^2 – ac)x – bc, we have a = ab, b = (b^2 – ac), and c = -bc.
Plugging these values into the quadratic formula, we get:
x = (-(b^2 – ac) ± √((b^2 – ac)^2 - 4ab(-bc))) / 2ab
Simplifying this expression will give us the zeroes of the polynomial.

Relationship between the zeroes and coefficients:
The relationship between the zeroes of a quadratic polynomial and its coefficients can be explained by Vieta's formulas.
Vieta's formulas state that for a quadratic equation ax^2 + bx + c = 0 with zeroes α and β, the sum of the zeroes α + β is equal to -b/a, and the product of the zeroes αβ is equal to c/a.
By verifying these relationships for the given polynomial, we can confirm the relationship between the zeroes and coefficients. This relationship provides insights into how the coefficients of a quadratic polynomial are related to its roots.