Unit+3+Virtual+Homework


 * __Unit 3 Lesson 5__**

Solve the following problems on a separate piece of paper. Use your solutions to answer the following summary questions in your virtual notebook.

a.) Divide f(x) by g(x) using synthetic or polynomial division.
 * Example 1:**


 * f(x) = x^3 - x^2 + 2x -1 g(x) = x + 3**

b.) Find f(-3) when f(x) = x^3 - x^2 + 2x -1

a.) Divide f(x) by g(x) using synthetic or polynomial division.
 * Example 2:**


 * f(x) = 2x^3 - 3x^2 + 4x -7 g(x) = x - 2**

b.) Find f(2) when f(x) = 2x^3 - 3x^2 + 4x - 7

a.) Divide f(x) by g(x) using synthetic or polynomial division.
 * Example 3:**


 * f(x) = 4x^3 - 9x^2 + 3x -10 g(x) = x + 2**

b.) Find f(-2) when f(x) = 4x^3 - 9x^2 + 3x -10


 * __Summary Questions__**
 * 1. What do you notice about your solutions in part a and part b. In your explanation include what you got for solution to parts a and b to support your explanation.**


 * 2. How can what you found be used as a short cut method to see if a number is a zero of a polynomial function or if a binomial is a factor before starting the synthetic division process? Explain.**


 * 3. Look up the definition of the remainder theorem and factor theorem on page 215 and 216 of your text. Explain what these theorems mean in your own words using the examples above. Are there any restrictions to using the remainder theorem? Explain.**


 * 4. Explain when polynomial division is the appropriate method to use when dividing two polynomials. Explain when synthetic division is the most appropriate method to be used. Can you divide f(x) = 4x^3 - 8x^2 + 2x - 1 by g(x) = 2x + 1 using synthetic division? If you can explain what you would use as your k value.**


 * __Unit 3 Lesson 9__**

1. The Fundamental Theorem of Algebra States: A polynomial function of a degree //n// has //n// zeros(real and non real). Some of these zeros may be repeated. Every polynomial of odd degree has at least one zero.


 * Explain what this statement means in your own words. In your description you should include an algebraic or graphical example to support your statement. You should also include the vocabulary of complex zeros, real zeros, and repeated zeros.**

2. Is it possible to find a polynomial with a degree of 3 with real number coefficients that has -2 as its only real zero? Explain.

3. The complex conjugate theorem states: Suppose that f(x) is a polynomial function with real coefficients. If a and b are real number with b not equal to zero and a + bi is a zero of f(x) then its complex conjugate a - bi is also a zero of f(x).


 * Explain what this statement means in your own words. You should include examples of complex conjugates when making your statement,.**

4. Is it possible to find a polynomial function of a degree of 4 with real coefficients that has zeros 1+3i and 1-i. Explain.

5. Is it possible to find a polynomial function of a degree of 4 with real coefficients that has zeros -3, 1 + 2i, and 1 - i. Explain.