math 110 ch. 4

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math 110 ch. 4
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2010-04-30 00:29:13
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all the different theorems
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  1. Remainder Theorem?
    If a polynomial f(x) is divided by x−c then the remainder is f(c).
  2. Factor Theorem ?
    A polynomial f(x) has a factor x−c if and only if f(c)=0.
  3. 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).
  4. maximum number of zeros theorem?
    A Polynomial of degree n>0 has at least n different roots (zeros)
  5. 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.
  6. 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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