What is the relationship between roots and discriminant of a polynomial?

The discriminant of a polynomial is related to the nature and number of its roots.

The discriminant of a quadratic polynomial ax^2 + bx + c is given by the expression b^2 - 4ac. It determines the nature of the roots of the quadratic equation. If the discriminant is positive, the quadratic has two distinct real roots. If the discriminant is zero, the quadratic has one real root of multiplicity 2. If the discriminant is negative, the quadratic has two complex conjugate roots.

For a polynomial of degree n, the discriminant is given by the expression Δ = (-1)^{n(n-1)/2} a_1^{2n-2} \prod_{i<j} (r_i - r_j)^2, where a_1 is the leading coefficient of the polynomial and r_1, r_2, ..., r_n are its roots. The discriminant is zero if and only if the polynomial has a multiple root.

If the discriminant is positive, the polynomial has n distinct real roots. If the discriminant is negative, the polynomial has n complex conjugate roots. If the discriminant is zero, the polynomial has at least one multiple root.

In summary, the discriminant of a polynomial is a useful tool for determining the nature and number of its roots. It is zero if and only if the polynomial has a multiple root, and positive or negative depending on whether the roots are real or complex conjugate.