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:
“Well: what we gain by science is, after all, sadness, as the Preacher saith. The more we know of the laws & nature of the Universe the more ghastly a business we perceive it all to be—& the non-necessity of it.”
—Thomas Hardy (1840–1928)
“The greatest gifts you can give your children are the roots of responsibility and the wings of independence.”
—Denis Waitly. Quoted in The Winning Family, ch. 25, by Louise Hart (1987)