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.
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