1. Homework, Page 223
Divide f (x) by d (x) and write a summary statement in polynomial form and fraction form 1.

2. Homework, Page 223
Divide f (x) by d (x) and write a summary statement in polynomial form and fraction form 5.

3. Homework, Page 223

4. Homework, Page 223
Divide using synthetic division, and write a summary statement in fraction form 9.

5. Homework, Page 223
Use the Remainder Theorem to find the remainder when f (x) is divided by x – k.
13.

6. Homework, Page 223
Use the Remainder Theorem to find the remainder when f (x) is divided by x – k.
17.

7. Homework, Page 223
Use the Factor Theorem to determine whether the first polynomial is a factor of the second polynomial.
21.
The first function is not a factor of the second because there is a remainder other than zero.

8. Homework, Page 223
Use the graph to guess possible factors of f (x). Then completely factor f (x) with the aid of synthetic division.
25.

9. Homework, Page 223
Find the polynomial function with the leading coefficient 2 that has the degree and zeroes.
29.

10. Homework, Page 223
Use the Rational Zero Theorem to write a list of all potential rational zeroes. Then determine, which if, any are zeroes.
33.

11. Homework, Page 223
Use synthetic division to prove that the number is an upper bound for the real zeroes of the function f.
37.

12. Homework, Page 223
Use synthetic division to prove that the number is a lower bound for the real zeroes of the function f.
41.

13. Homework, Page 223
Use the Upper and Lower Bound Tests to decide whether there could be real zeros for the function outside the window shown.
45.

14. Homework, Page 223
Find all real zeroes of the function, finding exact values when possible. Identify each zero as rational or irrational.

15. Homework, Page 223
Find all real zeroes of each function, finding exact values whenever possible. Identify each zero as rational or irrational.
53.

16. Homework, Page 223
The Sunspot Small Appliance Co. determines that the
supply function for their Evercurl hair dryer is S(p) = p3
and that its demand function is D(p) = 80 – 0.02p2 where p is
price. Determine the price for which supply equals demand and
the number of hair dryers corresponding to this equilibrium price.

17. Homework, Page 223 61. Let f (x) = x4 = 2x3 – 11x2 – 13x + 38.
a. Use the upper and lower bounds tests to show
that the real zeros all lie on the interval [–5, 4].
b. Find all of the rational zeros of f (x).
c. Factor f (x) using the rational zero(s) found in (b).
d. Approximate all the irrational zeros of f (x) .
e. Use synthetic division and the irrational zero(s) found in (d) to continue the factorization of f (x) begun in (c).

18. Homework, Page 223
61. a. Use the upper and lower bounds tests to show that the real zeros all lie on the interval [–5, 4].

19. Homework, Page 223 61. b. Find all of the rational zeros of f (x).
From the graph, the only possible rational root is x = 2.

20. Homework, Page 223
61. c. Factor using the rational zero(s) found in (b).
d. Approximate all the irrational zeros of f (x).

21. Homework, Page 223
61. e. Use synthetic division and the irrational zero(s) found in (d) to continue the factorization of f (x) begun in (c).

22. Homework, Page 223 65. Let f be a polynomial function with f(3) = 0.
Which of the following statements is not true?
a. x + 3 is a factor of f (x).
b. x – 3 is a factor of f (x).
c. x = 3 is a zero of f (x).
d. 3 is an x–intercept of f (x).
e. The remainder when f (x) is divided by x – 3 is zero.

23. Homework, Page 223
73. A classic theorem, Descartes’ Rule of Signs, tells us about the number of positive and negative real zeroes of a polynomial function by looking at the polynomial’s variations in sign. A variation in sign occurs when consecutive coefficients (in standard form) have opposite signs.
If f (x) = anxn + an–1 xn–1 + …. + ao is a polynomial of degree n, then
The number of positive real zeros of f is equal to the number of variations in sign of f (x) or that number less some even number.
The number of negative real zeros of f is equal to the number of variations in sign of f (–x) or that number less some even number.

24. Homework, Page 223
73. Use Descartes’ Rule of Signs to determine the possible number of positive and negative real zeroes of the function. a.

25. Homework, Page 223
73. Use Descartes’ Rule of Signs to determine the possible number of positive and negative real zeroes of the function. b.

26. Homework, Page 223
73. Use Descartes’ Rule of Signs to determine the possible number of positive and negative real zeroes of the function. c.

27. Homework, Page 223
73. Use Descartes’ Rule of Signs to determine the possible number of positive and negative real zeroes of the function. d.

28. Complex Zeros and the Fundamental Theorem of Algebra
2.5
Complex Zeros and the Fundamental Theorem of Algebra

29. What you’ll learn about
Two Major Theorems
Complex Conjugate Zeros
Factoring with Real Number Coefficients
… and why
These topics provide the complete story about
the zeros and factors of polynomials with real number
coefficients.

30. Fundamental Theorem of Algebra
A polynomial function of degree n has n complex zeros (real and imaginary). Some of these zeros may be repeated.

31. Linear Factorization Theorem

32. Fundamental Polynomial Connections in the Complex Case
The following statements about a polynomial function f are equivalent, if k is a complex number:
1. x = k is a solution (or root) of the equation f(x) = 0
2. k is a zero of the function f.
3. x – k is a factor of f(x).

33. Example Exploring Fundamental Polynomial Connections

34. Complex Conjugate Zeros

35. Example Finding a Polynomial from Given Zeros

36. Factors of a Polynomial with Real Coefficients
Every polynomial function with real coefficients can be written as a product of linear factors and irreducible quadratic factors, each with real coefficients.

37. Example Factoring a Polynomial

38. Polynomial Functions of Odd Degree

39. Homework
Homework Assignment #2
Read Section: 2.6
Page 234, Exercises: 1 – 61 (EOO)
Quiz next time

40. Graphs of Rational Functions
2.6
Graphs of Rational Functions

41. Quick Review

42. Quick Review Solutions

43. What you’ll learn about
Rational Functions
Transformations of the Reciprocal Function
Limits and Asymptotes
Analyzing Graphs of Rational Functions
… and why
Rational functions are used in calculus and in scientific
applications such as inverse proportions.

44. Rational Functions

45. We used limits to investigate continuity in Chapter 1
Limits may also be used
to investigate behavior near vertical asymptotes
to investigate behavior as functions approach positive or negative infinity, usually called end behavior

46. Transformations of the Reciprocal Function
The general form for a function is
In this equation, k indicates units of vertical translation and h indicates units of horizontal translation.
For instance,
indicates the reciprocal
function is translated 4
units left and 3 down.

47. Example Finding the Domain of a Rational Function

48. Graph a Rational Function

49. Graph a Rational Function (Cont’d)

50. Example Finding Asymptotes of Rational Functions

51. Example Graphing a Rational Function

52. Analyzing Graphs
Use limits to describe the behavior of the function

53. Finding Properties of Graphs
Find the intercepts, vertical asymptotes, end behavior asymptotes, and graph the function.