Intermediate Value Theorem

1. 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.

2. 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.

3. Example: Find a polynomial with rational coefficients with zeros 2, 1 + , and 1 – i.

4. 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).

5. 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.

6. 2) Use the Intermediate Value Theorem to find the 1 unit integral interval for each of the indicated number of zeros. a) One zero:

7. 2) Use the Intermediate Value Theorem to find the 1 unit integral interval for each of the indicated number of zeros. b) Four zeros:

8. 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. .

9. 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.