Transcript Intermediate Value Theorem (IVT)

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.