Multiplicity of a root

The multiplicity of a polynomial root is the number of times its corresponding linear factor divides the polynomial. A root of multiplicity greater than one is repeated.

Factor theorem: A polynomial has root a exactly when it is divisible by x − a. It connects each root to the linear factor whose repetitions define its multiplicity.
Polynomial: An expression formed from variables and coefficients using addition, subtraction, and nonnegative integer powers. Multiplicity describes how a polynomial factors around one of its roots.
Descartes' rule of signs: A rule bounding positive and negative real roots of a polynomial by its coefficient sign changes. Its root counts include multiplicity, so repeated roots affect the bounds.
Simple root: A root of multiplicity one. It contrasts with repeated roots and usually gives a graph crossing the axis.
Polynomial division: An algebraic procedure that divides one polynomial by another to produce a quotient and remainder. Repeated division by x − a tests how many times a root’s factor occurs.
Root of a polynomial: A value that makes a polynomial equal zero. Multiplicity refines the description of an individual polynomial root.
Newton's method: An iterative method for approximating solutions of equations using tangent lines. Repeated roots often make its convergence slower than for simple roots.
Root multiplicity: The number of times a root occurs in a polynomial’s factorization, counted over the chosen coefficient field. This broader phrasing highlights that multiplicity depends on the polynomial and its coefficient field.
Derivative: A function’s derivative gives its instantaneous rate of change and local linear approximation. Vanishing successive derivatives at a root help identify its multiplicity.
Linear factor: A degree-one polynomial factor, commonly written as x − a. The repeated linear factor is the object counted by a root’s multiplicity.

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