Unit 4: Modeling with polynomial & rational functions & equations

1. Ms. C. Taylor Common Core Math 3 Unit 4: Modeling with polynomial & rational functions & equations

2. Warm-Up • Factor the following

3. Polynomials • Degree-highest exponent • Leading Coefficient-coefficient of the first term. • Left Behavior- what the graph does on the left side as • Right Behavior- what the graph does on the right side as

4. Handout • Fill in the handout on Investigation of Polynomial Functions

5. Relative Maxima or Minima • Relative Maxima is when the graph of changes from increasing to decreasing. • Relative Minima is when the graph of changes from decreasing to increasing.

6. Higher Degree Polynomial Functions and Graphs • an is called the leading coefficient • n is the degree of the polynomial • a0 is called the constant term Polynomial Function A polynomial function of degree n in the variable x is a function defined by where each ai is real, an 0, and n is a whole number.

7. f(x) = 3 Polynomial Functions ConstantFunction Degree = 0 Maximum Number of Zeros: 0

8. Polynomial Functions f(x) = x + 2 LinearFunction Degree = 1 Maximum Number of Zeros: 1

9. Polynomial Functions f(x) = x2 + 3x + 2 QuadraticFunction Degree = 2 Maximum Number of Zeros: 2

10. Polynomial Functions f(x) = x3 + 4x2 + 2 Cubic Function Degree = 3 Maximum Number of Zeros: 3

11. Polynomial Functions Quartic Function Degree = 4 Maximum Number of Zeros: 4

12. As xincreases or decreases without bound, the graph of the polynomial function f (x)=anxn+ an-1xn-1+ an-2xn-2 +…+a1x + a0 (an¹ 0) eventually rises or falls. In particular, For n odd: an> 0 an< 0 If the leading coefficient is positive, the graph falls to the left and rises to the right. If the leading coefficient is negative, the graph rises to the left and falls to the right. Rises right Rises left Falls right Falls left The Leading Coefficient Test

13. As xincreases or decreases without bound, the graph of the polynomial function f (x)=anxn+ an-1xn-1+ an-2xn-2 +…+a1x + a0 (an¹ 0) eventually rises or falls. In particular, For n even: an> 0an< 0 If the leading coefficient is positive, the graph rises to the left and to the right. If the leading coefficient is negative, the graph falls to the left and to the right. Rises right Rises left Falls left Falls right The Leading Coefficient Test

14. Rises right y x Falls left Example Use the Leading Coefficient Test to determine the end behavior of the graph of f (x)= x3+3x2-x- 3.

15. Determining End Behavior Match each function with its graph. B. A. C. D.

16. x-Intercepts (Real Zeros) Number Of x-Intercepts of a Polynomial Function A polynomial function of degree n will have a maximum of nx- intercepts (real zeros). Find all zeros of f (x)=-x4+4x3 - 4x2. -x4+4x3- 4x2 = 0 We now have a polynomial equation. x4-4x3+ 4x2= 0 Multiply both sides by -1. (optional step) x2(x2- 4x+ 4) = 0 Factor out x2. x2(x- 2)2= 0 Factor completely. x2= 0 or (x- 2)2= 0 Set each factor equal to zero. x= 0 x= 2Solve for x. (0,0) (2,0)

17. Multiplicity and x-Intercepts If r is a zero of even multiplicity, then the graph touches the x-axis and turns around at r. If r is a zero of odd multiplicity, then the graph crosses the x-axis at r. Regardless of whether a zero is even or odd, graphs tend to flatten out at zeros with multiplicity greater than one.

18. Example • Find the x-intercepts and multiplicity of f(x) =2(x+2)2(x-3) • Zeros are at (-2,0) (3,0)

19. Extrema • Turning points – where the graph of a function changes from increasing to decreasing or vice versa. The number of turning points of the graph of a polynomial function of degree n  1 is at most n – 1. • Local maximum point – highest point or “peak” in an interval • function values at these points are called local maxima • Local minimum point – lowest point or “valley” in an interval • function values at these points are called local minima • Extrema – plural of extremum, includes all local maxima and local minima

20. Extrema

21. Warm-Up • Given the following polynomial , state how many zeros are possible, left behavior, right behavior, and name the polynomial.

22. End Behavior

23. Domain & Range • It depends on the polynomial graph that you are looking at. • Most likely the domain will be • Most likely the range will be

24. Effects of • The “h” will shift the graph left or right depending on sign • The “k” will shift the graph up or down depending on the sign.

25. Remainder Theorem • If I divide the polynomial by the factor given then I will have a remainder other than zero. • If you plug in the constant of the factor into the equation then you should come up with the remainder when doing synthetic division.

26. Examples-Remainder Theorem

27. Warm-Up • Determine the remainder of the following:

28. Factor Theorem • In order to prove that (x-a) is a factor of the polynomial f(a)=0. • In other words, the remainder from synthetic division has to be zero.

29. Examples

30. Fundamental Theorem of Algebra • Every polynomial equation with degree greater than zero has at least one root in the set of complex numbers.

31. Rational Zero Theorem • If is a rational number in simplest form and is a zero of the polynomial then p is a factor of and q is a factor of .

32. Examples

33. Warm-Up • Determine if the factor is a factor of • State all of the possible rational zeros for the following: .

34. Rational Equations • Step 1: Set all denominators equal to zero and solve. • Step 2: Find the CD (Factor Denom.) • Step 3: Look for “what is missing” & multiply each fraction by the appropriate expression. DUMP the denominators. • Step 4: Solve for x. • Step 5: Check to be sure that x doesn’t equal any number from step 1.

35. Rational Equations • Examples:

36. Warm-Up • Solve the following:

37. Literal Equations • Solve the following for the specified variable • , solve for h • , solve for c • , solve for r

38. Warm-Up • Solve the following for the indicated variable:

39. Practice • An open box is to be made from a rectangular piece of material 20 inches by 12 inches by cutting equal squares from the corners and turning up the sides. • What is the maximum volume that the box can hold?

40. Practice • An open box is to be made from a rectangular piece of material 15 inches by 10 inches by cutting equal squares from the corners and turning up the sides. • What is the maximum volume that the box can hold?

41. Practice • An open box is to be made from a rectangular piece of material 6inches by 1 inch by cutting equal squares from the corners and turning up the sides. • What is the maximum volume that the box can hold?

42. Practice • An open box is to be made from a rectangular piece of material 25 inches by 14 inches by cutting equal squares from the corners and turning up the sides. • What is the maximum volume that the box can hold?

43. Practice • An open box is to be made from a rectangular piece of material 35 inches by 22 inches by cutting equal squares from the corners and turning up the sides. • What is the maximum volume that the box can hold?

44. Practice • An open box is to be made from a rectangular piece of material 40 inches by 24 inches by cutting equal squares from the corners and turning up the sides. • What is the maximum volume that the box can hold?

45. Warm-Up • Write the following polynomial in standard form and give the degree, name of the polynomial, leading coefficient.

46. Finding k in a Polynomial • Substitute the value from the factor in for x and then set the equation equal to zero. • Check by using synthetic division to make sure that you get a remainder of 0.

47. Examples • given (x-2) is a factor • given (x-2) is a factor

48. Practice • given is a factor • given is a factor • given is a factor • given is a factor

49. Warm-Up • If is a factor of , then find the value of k.

50. To write the polynomial equation given the zeros: Step 1: Rewrite the zeros as factors (x – ( )) (x – ( )) (x – ( ))