Zeros of Polynomial Functions Section 2.5. Objectives Use the Factor Theorem to show that x-c is a factor a polynomial. Find all real zeros of a polynomial.

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Presentation transcript:

Zeros of Polynomial Functions Section 2.5

Objectives Use the Factor Theorem to show that x-c is a factor a polynomial. Find all real zeros of a polynomial given one or more zeros. Find all the rational zeros of a polynomial using the Rational Zero Test. Find all real zeros of a polynomial using the Rational Zero Test. Find all zeros of a polynomial. Write the equation of a polynomial given some of its zeros.

Vocabulary rational zero real zero multiplicity

Factor Theorem Let f (x) be a polynomial a.If f(c) = 0, then x – c is a factor of f (x). b.If x – c, is a factor of f(x), then f(c) = 0.

If c = 3 is a zero of the polynomial find all other zeros of P(x).

Use synthetic division to show that x = 6 is a solutions of the equation

Rational Root (Zero) Theorem (Test) If has integer coefficients and (where is reduced to lowest terms) is a rational zero of f, then p is a factor of the constant term, a 0, and q is a factor of the leading coefficient, a n.

Find all the rational zeros of the polynomial

Find all the real zeros of the polynomial

If, where n ≥ 1 and a n ≠ 0, then Where c 1, c 2,..., c n are complex numbers (possibly real and not necessarily distinct). Linear Factorization Theorem

Factor into linear and irreducible quadratic factors with real coefficients.

Find all the zeros of the polynomial

Find the equation of a polynomial of degree 4 with integer coefficients and leading coefficient 1 that had zeros x = -2-3i, and at x = 1 with x = 1 a zero of multiplicity 2.

Let, Be a polynomial with real coefficients. 1.The number of positive real zeros of f is either a.the same as the number of sign changes of f(x) OR b.less than the number of sign changes of f(x) by a positive even integer. Descartes’s Rule of Signs If f(x) has only one variation in sign, then f has exactly one positive real zero.

Let, Be a polynomial with real coefficients. 1.The number of negative real zeros of f is either a.the same as the number of sign changes of f(—x) OR b.less than the number of sign changes of f(—x) by a positive even integer. Descartes’s Rule of Signs If f(—x) has only one variation in sign, then f has exactly one negative real zero.

If f(x) is a polynomial of degree n, where n ≥ 1, then the equation f(x) = 0 has at least one complex root. Fundamental Theorem of Algebra