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Intermediate Value Theorem Objective: Be able to find complex zeros using the complex zero theorem & be able to locate values using the IVT TS: Explicitly assess information and draw conclusions Warm Up: Refresh your memory on what the complex zero theorem says then use it to answer the example question. • Complex Root Theorem: Given a polynomial function, f, if a + bi is a root of the polynomial then a – bi must also be a root. Example: Find a polynomial with rational coefficients with zeros 2, 1 + 3 , and 1 – i. Intermediate Value Theorem (IVT): Given real numbers a & b where a < b. If a polynomial function, f, is such that f(a) ≠ f(b) then in the interval [a, b] f takes on every value between f(a) to f(b). 1) First use your calculator to find the zeros of f ( x) 3 x 4 x3 2 x 2 5 8 Now verify the 1 unit integral interval that the zeros are in using the Intermediate Value Theorem. 2) Use the Intermediate Value Theorem to find the 1 unit integral interval for each of the indicated number of zeros. 3 2 g ( x ) 3 x 4 x x 3 a) One zero: 2) Use the Intermediate Value Theorem to find the 1 unit integral interval for each of the indicated number of zeros. b) Four zeros: f ( x) x 4 10 x 2 2 3) Given : f ( x) 3x3 4 x 2 3x 2 a) What is a value guaranteed to be between f(2) and f(3). b) What is another value guaranteed to be there? c) What is a value that is NOT guaranteed to be there? d) But could your value for c be there? Sketch a graph to demonstrate your answer. . 4) Given a polynomial, g, where g(0) = -5 and g(3) = 15: a) True or False: There must be at least one zero to the polynomial. Explain. b) True or False: There must be an x value between 0 and 3 such that g(x) = 12. Explain. c) True or False: There can not be a value, c, between 0 and 3 such that g(c) = 25. Explain.