Intermediate Value TheoremObjective: 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.3

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.

4 3 23( ) 2 58f x x x x

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

a) One zero: 3 2( ) 3 4 3g x x x x

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

b) Four zeros: 4 2( ) 10 2f x x x

3) Given :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.

.

3 2( ) 3 4 3 2f x x x x

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.