Unit+3+Virtual+Notebook


 * __Unit 2 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^2-4x+14)+(-43)
 * 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 f(x) = -43 a.) Divide f(x) by g(x) using synthetic or polynomial division.
 * Example 2: **

f(x) = (x-2)(2x^2+x+6)+(5)
 * 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 f(x) = 5 a.) Divide f(x) by g(x) using synthetic or polynomial division.
 * Example 3: **

f(x) = (x+2)(4x^2-17x +37)+(-84)
 * 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 f(x) = -84

All the answers I got for part A were the remainders for the answers of part B. For example, in example 1A the remainder was -43 and in example 1B the answer was -43.
 * __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. **

It can be used as a short cut method to see if a number is a zero of a polynomial function because you can plug in the numbers we got in the polynomial.
 * 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. **

Remainder Theorem - Used in polynomial long divison --> finding zeros Factor Theorem - Used in synthetic division --> finding zeros
 * 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. **

Polynomial division appropriate - divisor has a degree or coefficent Synthetic division appropriate - divisor is x-k or x+k
 * 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 2 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. The degree tells you the maximum amount of zeros in the function. A zero will be repeated if its multiplicity is more than one. Odd degrees cross the x-axis, the other zeros would be complex/imaginary f(x)= x^3+5x+3 Degree = 3 [3 zeros max] Odd = cross x-axis


 * 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. Yes, because there also could be complex numbers that equal zero.

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).

If a + bi is a complex number, then its complex conjugates, a - bi, must be its other zero.
 * 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. Yes, because the max number of zeros is 4 which could be 1+3i, 1-3i, 1-i, 1+i

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. No, because the max number of zeros would only be 4. If it has a zero of -3, 1+2i and 1-i the complex conjugate would also have 1+i and 1-2i be a zero and that is pass 4.