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Understand the concepts behind complex zeros, the Intermediate Value Theorem, and their applications in finding values within intervals. Practice using calculators and theorems to locate zeros and intervals accurately, ensuring a clear grasp of these mathematical principles.
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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 + , 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 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. 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:
3) Given : • What is a value guaranteed to be between f(2) and f(3). • What is another value guaranteed to be there? • What is a value that is NOT guaranteed to be there? • 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: • True or False: There must be at least one zero to the polynomial. Explain. • True or False: There must be an x value between 0 and 3 such that g(x) = 12. Explain. • True or False: There can not be a value, c, between 0 and 3 such that g(c) = 25. Explain.
