math 110 ch. 4
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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).
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