Multiplicity of A Root of A Polynomial
Let F be a field and p(x) be a polynomial in one variable and coefficients in F. An element a ∈ F is called a root of multiplicity k of p(x) if there is a polynomial s(x) such that s(a) ≠ 0 and p(x) = (x − a)ks(x). If k = 1, then a is called a simple root.
For instance, the polynomial p(x) = x3 + 2x2 − 7x + 4 has 1 and −4 as roots, and can be written as p(x) = (x + 4)(x − 1)2. This means that 1 is a root of multiplicity 2, and −4 is a 'simple' root (of multiplicity 1). Multiplicity can be thought of as "How many times does the solution appear in the original equation?".
The discriminant of a polynomial is zero if and only if the polynomial has a multiple root.
Read more about this topic: Multiplicity (mathematics)
Famous quotes containing the words multiplicity of, multiplicity and/or root:
“One might get the impression that I recommend a new methodology which replaces induction by counterinduction and uses a multiplicity of theories, metaphysical views, fairy tales, instead of the customary pair theory/observation. This impression would certainly be mistaken. My intention is not to replace one set of general rules by another such set: my intention is rather to convince the reader that all methodologies, even the most obvious ones, have their limits.”
—Paul Feyerabend (1924–1994)
“The point of cities is multiplicity of choice.”
—Jane Jacobs (b. 1916)
“Flower in the crannied wall,
I pluck you out of the crannies,
I hold you here, root and all, in my hand,
Little flower—but if I could understand
What you are, root and all, and all in all,
I should know what God and man is.”
—Alfred Tennyson (1809–1892)