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MATH 1100 SECTION 5.4 Notes The Fundamental Theorem of Algebra – Text Pages 346-352 We have already seen that an nth degree polynomial can have at most n real zeros. In the complex number system, an nth degree polynomial has exactly n zeros, so it can be factored into exactly n linear factors. The Fundamental Theorem of Algebra: Every polynomial Px a n x n a n 1 x n 1 ... a1 x a0 complex coefficients has at least one complex zero. n 1, an 0 with Complete Factorization Theorem: If P(x) is a polynomial of degree n>o, then there exist complex numbers a, c1 , c2 , ... , cn (with a 0 ) such that Px ax c1 x c2 x cn . Zeros Theorem: Every polynomial of degree n 1 has exactly n zeros, provided that a zero of multiplicity k is counted k times. Conjugate Zeros Theorem: If the polynomial P has real coefficients, and if the complex number z is a zero of P, then its complex conjugate, z is also a zero P. Linear and Quadratic Factors Theorem: Every polynomial with real coefficients can be factored into a product of linear and irreducible quadratic factors with real coefficients. Example 1: Factor the polynomial completely and find all its real and complex zeros. State the multiplicity of each zero. Px x 3 8x 2 9 x 28 Example 2: Find a polynomial with integer coefficients that satisfies the following: (a.) P has degree 3 and zeros 3, 2i, and –2i, with P2 32 . (b.) P has degree 3 and zeros –3, and 2+4i. Example 3: For the polynomial Px x 4 8 x 2 9 (a.) Factor P into linear and irreducible quadratic factors with real coefficients. (b.) Factor P completely into linear factors with complex coefficients.