Section 4.3Zeros of Polynomials

Approximate the Zeros

Approximate the Zeros

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 a0

The denominator, d, of the zero is a factor of the leading coefficient an.

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