Nature of The Roots
The discriminant gives additional information on the nature of the roots beyond simply whether there are any repeated roots: it also gives information on whether the roots are real or complex, and rational or irrational. More formally, it gives information on whether the roots are in the field over which the polynomial is defined, or are in an extension field, and hence whether the polynomial factors over the field of coefficients. This is most transparent and easily stated for quadratic and cubic polynomials; for polynomials of degree 4 or higher this is more difficult to state.
Read more about this topic: Discriminant
Famous quotes containing the words nature and/or roots:
“Habit is a second nature that destroys the first. But what is nature? Why is habit not natural? I am very much afraid that nature itself is only a first habit, just as habit is a second nature.”
—Blaise Pascal (1623–1662)
“Where the world ends
The mind is made unchanging, for it finds
Miracle, ecstasy, the impossible hope,
The flagstone under all, the fire of fires,
The roots of the world.”
—William Butler Yeats (1865–1939)