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SWBAT use Descartes’s Rule of Signs to find zeros of polynomials and write polynomial functions given the zeros (Section 3-5) Warm up List all the possible rational zeros of f, and determine all real zeros of f. 1 Another Test for Zeros of Polynomials Theorem: Descartes’ Rules of Signs Let f(x) = anxn + an-1xn-1 + …..+a1x + a0 be a polynomial with real coefficients and a0 ≠ 0. a) The number of positive real zeros of f is either equal to the number of variations in sign of f(x) or less than that number by an even integer. b) The number of negative real zeros of f is either equal to the numbers of variations in sign of f(-x) or less than that number by an even integer. ** A variation in sign means that two consecutive coefficients have opposite signs. The rule provides information about the number and location of the real zeros of a polynomial function. Example 1) Using Descartes’s rule of signs describe the possible real zeros of a) g(x) = 3x 3 – 5x2 + 6x – 4 b) f(x) = 5x 5 +10x 2 c) g(x) = 2x 3 – 3x2 – 3 3 Example 2) Find the real and complex zeros of f ( x) 6 x 3 4 x 2 3x 2 4 Extra Practice Using Descartes’s rule of signs describe the possible real zeros of 1) f ( x) 3x 3 2 x 2 x 3 3 2 2) g ( x) 2 x 3x 3 3) Factor completely and find all the real and complex zeros. P(x) = x 3 + x2 – 4x – 4 5 4) ) Factor completely and find all the real and complex zeros. 5) P(x) = x4 - 5x 3 + 3x2 + x 6 6) ) P(x) = 3x 6 – 4x4 + 3x3 + 2x2 – x - 3 7