math 110 ch. 4
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Remainder Theorem?
If a polynomial f(x) is divided by x−c then the remainder is f(c).

Factor Theorem ?
A polynomial f(x) has a factor x−c if and only if f(c)=0.

Complete Factorization Theorem for Polynomials ?
 If f(x) is a polynomial of degree n>0, then there exist n complex numbers c
 1, c2, c3, ... , cn such
 that f(x)=a(x−c 1)(x−c2)⋅⋅⋅(x−cn), where a is the leading coefficient of f(x). Each number ck is
 a zero of f(x).

maximum number of zeros theorem?
A Polynomial of degree n>0 has at least n different roots (zeros)

Descartes’ Rule of Signs ?
 Let f(x) be a polynomial with real coefficients and a nonzero constant term.
 Step 1: The number of positive real zeros of f(x) either is equal to the number of variations of sign in
 f(x) or is less than that number by an even integer.
 Step 2: The number of negative real zeros of f(x) either is equal to the number of variations of sign in
 f(−x) or is less than that number by an even integer.

Definition of Multiplicity?
 If a factor x−c occurs m times in the factorization of f(x), then c is a zero
 of multiplicity m of the polynomial f(x).