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Section 8.4: Roots and Zeros Date____ Fundamental Theorem of Algebra: Every polynomial equation with degree greater than zero has at least one root in the set of complex numbers (real or imaginary) Corollary: A polynomial equation with degree n and with complex coefficients has exactly n roots in the set of complex numbers Date____ Examples: 1. Find all roots of 0 = x3 + x2 - 4x - 4 f(x) = 0 means x - intercepts Strategies: 1. graph it to look for real zeros -table -trace -2nd calc (zeros) 2. Use synthetic division to get to "x2" term 3. Solve the quadratic (factor, QF) Complex Conjugate Theorem: Suppose a and b are real numbers with b ≠ 0. If a + bi is a zero of a polynomial function, then a - bi is also a zero of the function. 2. Given a function and one of its zeros, find the remaining zeros of the function. a. f(x) = x3 - 7x2 + 16x - 10; 3 - i graph on calculator Strategies: 1. graph it to look for real zeros -table -trace -2nd calc (zeros) 2. Use synthetic division to get to "x2" term 3. Solve the quadratic (factor, QF) b. g(x) = x3 - 2x2 + 5x + 26; -2 Descartes' Rule of Signs: If we have a polynomial in descending order then: - the number of positive real zeros of the polynomial is the same as the number of changes in sign of the coefficient of the terms, or less than this by an even number, and - the number of negative real zeros of the polynomial is the same as the number of changes in sign of the coefficient of the terms of p(-x), or less than this by an even number. 3. State the number of positive real zeros, negative real zeros, and imaginary zeros for a. b. g(x) = -9x3 - x2 + 5x - 8 g(x) = x5 - x4 + x3 + x - 7 4. Write the polynomial function of least degree with integer coefficients whose zeros include the following. a. 1 and 1 + i b. -2 and 2 + 3i