Section 4.3 Zeros of Polynomials. Approximate the Zeros.

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Presentation transcript:

Section 4.3 Zeros of Polynomials

Approximate the Zeros

Fundamental Theorem of Algebra  If a polynomial f(x) has positive degree and complex coefficients, then f(x) has at least one complex zero.

Descartes’ Rule of Signs  Let f(x) be a polynomial with real coefficients and a nonzero constant term.  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  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.

Rational Root Theorem  If the polynomial  has integer coefficients and if c/d is a rational zero of f(x) such that c and d have no common prime factor, then:  The numerator, c, of the zero is a factor of the constant term a 0  The denominator, d, of the zero is a factor of the leading coefficient a n.