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 of the, nature of, nature and/or roots:
“Religion. A daughter of Hope and Fear, explaining to Ignorance the nature of the Unknowable.”
—Ambrose Bierce (1842–1914)
“Because it is in the nature of things that they become extreme, we have passed down from manliness to cruelty. If I had been told when I was 20 that there was a tavern in the town where the brave and the cruel were gathered together, I would have run all the way and I would have gone up to the largest and leatheriest of the denizens and said: “If you truly love me, kill the bartender.””
—Quentin Crisp (b. 1908)
“To eat steak rare ... represents both a nature and a morality.”
—Roland Barthes (1915–1980)
“Sensuality often accelerates the growth of love so much that its roots remain weak and are easily pulled up.”
—Friedrich Nietzsche (1844–1900)