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:
“Is a Bill of Rights a security for [religious liberty]? If there were but one sect in America, a Bill of Rights would be a small protection for liberty.... Freedom derives from a multiplicity of sects, which pervade America, and which is the best and only security for religious liberty in any society. For where there is such a variety of sects, there cannot be a majority of any one sect to oppress and persecute the rest.”
—James Madison (17511836)
“The point of cities is multiplicity of choice.”
—Jane Jacobs (b. 1916)
“I will go root away
The noisome weeds which without profit suck
The soils fertility from wholesome flowers.”
—William Shakespeare (15641616)