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Mathematics LibreTexts

3.3e: Exercises - Polynomial End Behaviour

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A: Concepts

Exercise 3.3e.A 

1) Explain the difference between the coefficient of a power function and its degree.

2) If a polynomial function is in factored form, what would be a good first step in order to determine the degree of the function?

3) In general, explain the end behavior of a power function with odd degree if the leading coefficient is positive.

4) What can we conclude if, in general, the graph of a polynomial function exhibits the following end behavior? As x,f(x) and as x,f(x).

Answers to odd exercises:

1. The coefficient of the power function is the real number that is multiplied by the variable raised to a power. The degree is the highest power appearing in the function.

3. As x decreases without bound, so does f(x). As x increases without bound, so does f(x).

B: Identify Power Functions and Polynomials

Exercise 3.3e.B 

 Identify the function as a power function, a polynomial function, or neither. If neither, explain.

5) f(x)=(x2)3

6) f(x)=x5

7) f(x)=xx4

8) f(x)=x2x21

9) f(x)=3x+1

10) f(x)=2x(x+2)(x1)2

Answers to odd exercises:

5. Power function, 7. Polynomial,  9. Neither - this is an exponential function because the variable is in the exponent

C: Degree and Leading Coefficient of a Polynomial

Exercise 3.3e.C 

(a) Find the degree and leading coefficient for the given polynomial. (b) State the end behaviour.

11) 72x2

12) 3x4

13) x(4x2)(2x+1)

14) 2x23x5+x6

15) x2(2x3)2

16) 3(2x+5)2(1x)3

Answers to odd exercises:
11. Degree 2, Coefficient = 2  13. Degree 4, Coefficient = 2  15. Degree 4, Coefficient = 4  or 

D: Polynomial End Behaviour

Exercise 3.3e.D 

 Determine the end behavior of the functions.

17) f(x)=x4

18) f(x)=x3

19) f(x)=x4

20) f(x)=x9

 

21) f(x)=2x43x2+x1

22) f(x)=3x2+x2

23) f(x)=x2(2x3x+1)

24(a) f(x)=(2x)7

24(b) f(x)=3x(2x3)4

25) f(x)=x45x2

26) f(x)=x3

27) f(x)=(x1)(x2)(3x)

28) f(x)=x2(1x)2

29) f(x)=x510x4

Answers to odd exercises:

17. As x±, f(x), or 

19. As x±, f(x), or 

21. As x±, f(x), or 

23. As x, f(x), as x, f(x)
or 

25. As x±, f(x), or 

27. as x, f(x), as x, f(x)
or 

29.  As x, f(x), as x, f(x)
or 

 

E: Polynomial Intercepts

Exercise 3.3e.E 

 Find the intercepts of the functions.

30) C(t)=2(t4)(t+1)(t6)

31) f(t)=2(t1)(t+2)(t3)

32) g(n)=2(3n1)(2n+1)

33) f(x)=x416

34) f(x)=x3+27

35) f(x)=x(x22x8)

36) f(x)=(x+3)(4x21)

37) C(t)=3(t+2)(t3)(t+5)

38) C(t)=4t(t2)2(t+1)

39) C(t)=2t(t3)(t+1)2

40) C(t)=2t48t3+6t2

41) C(t)=4t4+12t340t2

42) f(x)=x4x2

43) f(x)=x3+x220x

44) f(x)=x3+6x27x

45) f(x)=x3+x24x4

46) f(x)=x3+2x29x18

47) f(x)=2x3x28x+4

48) f(x)=x67x38

49) f(x)=2x4+6x28

50) f(x)=x33x2x+3

51) f(x)=x62x43x2

52) f(x)=x63x44x2

53) f(x)=x55x3+4x

 Determine the intercepts and the end behavior.

55) f(x)=x3(x2)

56) f(x)=x(x3)(x+3)

57) f(x)=x(142x)(102x)

58) f(x)=x(142x)(102x)2

59) f(x)=x316x

60) f(x)=x327

61) f(x)=x481

62) f(x)=x3+x2+2x

63) f(x)=x32x215x

64) f(x)=x30.01x

Answers to odd exercises:

31.(0,12); (1,0),(2,0),(3,0)

33. (0,16); (2,0),(2,0).

35. (0,0),(4,0),(2,0).

37. (0,90); (2,0),(3,0),(5,0)

39. (0,0), (3,0),(1,0)

41. (0,0),(5,0),(2,0)

43. (0,0),(5,0),(4,0)

45. (0,4); (2,0),(2,0),(1,0)

47. (0,4)(2,0),(2,0),(12,0)

49. (0,8); (1,0),(1,0)

51. (0,0),(3,0)),(3,0)

53. (0,0),(1,0),(1,0),(2,0),(2,0)

55. (0,0); (0,0),(2,0).  As x±, f(x)

57. (0,0)(0,0),(5,0),(7,0). As x, f(x), as x, f(x)

59. (0,0); (4,0),(0,0),(4,0). As x, f(x), as x, f(x)

61. (0,81); (3,0),(3,0). As x±, f(x)

63.(0,0); (3,0),(0,0),(5,0). As x, f(x), as x, f(x)


3.3e: Exercises - Polynomial End Behaviour is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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