Unit+3+Virtual+Notebook

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. I noticed that each time I solved parts a, it never turns out perfectly, however, in part b, when f(x) is equal to the number that is doing the divisions, like in 1, it was -3, and in number 2, it was 2, and for 3, it was -2, the remainder of the long division or synthetic division was always the answer to the f(x) in parts b, for example, in example 1, the remainder was -43, which was also the answer for f(-3). 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. i can use this information to find if a number is a zero of a polynomial because if the remainder equals to the number being used in the synthetic equation, then there is no zero. 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. Remainder Theorem- it is a way in which to find the polynomial using long division. It helps to find whether the zeros. it can be used by long division, such as in example one. the remainder is -43, so it does not fit evenly into the equation. Factor Theorem- it uses the synthetic division method, and is used exactly for the same purpose as the Remainder Theorem. 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. You use the long division when the exponent of the leading coefficient of the divisor is larger than one. on the other hand, if the exponent is 1, then synthetic division can be used. As for the f(X), you can divide it by 2x+1 because the degree of the leading coefficient is 1, therefore, the k, which is the value of x if you were trying to solve for it in 2x+1, would help to find the polynomial, which can be used to find the other zeros.
 * __Unit 2 Lesson 5__**
 * __Summary Questions__**

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 might be repeated if its multiplicity is more than one. A function with an odd degree will has to have crossed the x-axis at least once, revealing at least one real zero, while the other zeros are complex zeros, or imaginary numbers. f(x) = x^3 + x^2 + x + 1 Degree: 3 ; will have, at most, 3 zeros Odd ; will cross the x-axis at least once, leaving at least one real zero. 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, an odd degree function may have one or more real zeros, meaning that -2 could be its only real zero while the others are complex zeros. 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). In a polynomial function, if one of its zeros is a complex number, a + bi, then its complex conjugates, a - bi, must also be its zero. 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 this function could have a total of 4 zeros, which could be 1 + 3i, its complex conjugates, 1 - 3i, 1 - i, and its complex conjugates, 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 since this function could only have a maximum total of 4 zeros, 1 + 2i and its complex conjugates, 1 - 2i, and 1 - i and its complex conjugates, 1 + i, should be the function's zeros. This is not the case since -3 is also a zero, making the total number of zeros to be 5.
 * __Unit 2 Lesson 9__**
 * 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.**
 * Explain what this statement means in your own words. You should include examples of complex conjugates when making your statement,.**