Multiple Root

A multiple root of the polynomial

f(x) = a_oxⁿ + a₁x^n-1 + … + a_n

is a number c such that f(x) is divided without remainder by the binomial (x—c) raised to the second or higher degree; c is called a root of multiplicity k iff(x) is divided by (x—c)^k but not by (x—c)^k+l. A root of multiplicity k of the polynomial f(x) is also a root of all the derivatives of the polynomial up to and including the derivative of order (k — 1), that is, of the polynomials f’(x), f”(x), & ,f^(k-1)(x). A multiple root of the polynomial f(x) is also called a multiple root of the equation f(x) = 0.