discriminant
Sign in to saveIn mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the original polynomial. The discriminant is widely used in polynomial factoring, number theory, and algebraic geometry.
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Article
26 sectionsContents
- Origin
- Definition
- Expression in terms of the roots
- Low degrees
- Degree 2
- Degree 3
- Degree 4
- Properties
- Zero discriminant
- Invariance under change of the variable
- Invariance under ring homomorphisms
- Product of polynomials
- Homogeneity
- Real roots
- Homogeneous bivariate polynomial
- Use in algebraic geometry
- Generalizations
- Quadratic forms
- Conic sections
- Real quadric surfaces
- Discriminant of an algebraic number field
- Fundamental discriminants
- Quadratic number fields
- Prime factorization
- References
- External links
In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the original polynomial. The discriminant is widely used in polynomial factoring, number theory, and algebraic geometry.
The discriminant of the quadratic polynomial ax^2+bx+c is b^2-4ac, the quantity which appears under the square root in the quadratic formula. If a\ne 0, this discriminant is zero if and only if the polynomial has a double root. In the case of real coefficients, it is positive if the polynomial has two distinct real roots, and negative if it has two distinct complex conjugate roots. Similarly, the discriminant of a cubic polynomial is zero if and only if the polynomial has a multiple root. In the case of a cubic with real coefficients, the discriminant is positive if the polynomial has three distinct real roots, and negative if it has one real root and two distinct complex conjugate roots.