Intermediate Value Theorem: Given real numbers a & b where ...



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 + [pic], 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).

Examples:

1) First use your calculator to find the zeros of [pic].

Now verify the1 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: [pic]

b. Four zeros: [pic]

3) Given [pic]:

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.

Homework:

1) Find the equation of the quadratic whose only x-intercept is (-4, 0) and passes through (-2, 8)

2) Graph y = -½ x2 + ½ x + 3.

3) Given [pic]

a. What is the end behavior

b. List the possible rational roots

4) Divide: [pic]

5) Use the remainder theorem to find f(5) for f(x) from problem #3.

6) Use your calculator to approximate the real zeros and relative extrema of the following functions.

a. [pic]

b. [pic]

7) Find a polynomial function that has the given zeros.

a. 0, 4

b. 0, -2, -3

c. 4, -3, 3, 0

d. 1 + [pic]

e. 2, 4 –[pic]

8) For each of the following: (a) use the IVT to find integral intervals one in length which must contain a zero (b) now use your calculator to find the zeros (checking your answer to part (a).

a. [pic]

b. [pic]

9) For each of the following: Identify the symmetry it has (x-axis, y-axis or origin), and determine the number of x-intercepts it has. Use your calculator to verify your answer.

a. [pic]

b. [pic]

c. [pic]

d. [pic]

10) An open box is to be made from a square piece of material 36cm on a side by cutting equal squares with sides of length x from the corners and turning up the sides.

a. Find an equation for the volume of the box, V(x)

b. Determine the domain of the function V

c. Use your calculator to find the length, x, for which the maximum volume is produced.

Answers: 1) y = 2x2 + 16x + 32 2) check with calc 3) a) up, up b)[pic]

4) [pic] 5) 2688 6) a) [pic]

b) [pic] 7) a) [pic] b) [pic]

c) [pic] d) [pic] e) [pic]

8) a) (-1,0) (1, 2) (2, 3): -0.879, 1.347, 2.532 b) (-2, -1) (0,1): -1.585, 0.779

9) check with calc 10) a) V(x) = x(36 – 2x)2 b) Domain (0, 18) c) 6cm

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