B Apply

For each equation, state the number of complex roots, the possible number of real roots, and the possible rational roots.

26. 2 x 4 − x 3 + 2 x 2 + 5 x − 26 = 0 2 x to the fourth , minus , x cubed , plus 2 , x squared , plus 5 x minus 26 equals 0
27. x 5 − x 3 − 11 x 2 + 9 x + 18 = 0 x to the fifth , minus , x cubed , minus , 11 x squared , plus 9 x plus 18 equals 0
28. − 12 + x + 10 x 2 + 3 x 3 = 0 negative 12 plus x plus , 10 x squared , plus , 3 x cubed , equals 0
29. 4 x 6 − x 5 − 24 = 0 4 x to the sixth , minus , x to the fifth , minus 24 equals 0

Find all the zeros of each function.

30. y = x 3 − 4 x 2 + 9 x − 36 y equals , x cubed , minus , 4 x squared , plus 9 x minus 36
31. f ( x ) = x 3 + 2 x 2 − 5 x − 10 f open x close equals . x cubed , plus 2 , x squared , minus 5 x minus 10
32. y = 2 x 3 − 3 x 2 − 18 x − 8 y equals , 2 x cubed , minus , 3 x squared , minus 18 x minus 8
33. y = 3 x 3 − 7 x 2 − 14 x + 24 y equals , 3 x cubed , minus , 7 x squared , minus 14 x plus 24
34. g ( x ) = x 3 − 4 x 2 − x + 22 g open x close equals . x cubed , minus , 4 x squared , minus x plus 22
35. y = x 3 − x 2 − 3 x − 9 y equals , x cubed , minus , x squared , minus 3 x minus 9
36. y = x 4 − x 3 − 5 x 2 − x − 6 y equals , x to the fourth , minus , x cubed , minus , 5 x squared , minus x minus 6
37. y = 2 x 4 + 3 x 3 − 17 x 2 − 27 x − 9 y equals , 2 x to the fourth , plus , 3 x cubed , minus , 17 x squared , minus 27 x minus 9

38. Think About a Plan A polynomial function, f ( x ) = x 4 − 5 x 3 − 28 x 2 + 188 x − 240 , f open x close equals . x to the fourth , minus , 5 x cubed , minus , 28 x squared , plus 188 x minus 240 comma  is used to model a new roller coaster section. The loading zone will be placed at one of the zeros. The function has a zero at 5. What are the possible locations for the loading zone?

    • Can you determine how many zeros you need to find?
    • How can you use polynomial division?
    • What other methods can be helpful?
39. Bridges A twist in a river can be modeled by the function f ( x ) = 1 3 x 3 + 1 2 x 2 − x , − 3 ≤ x ≤ 2 . f , open x close , equals , 1 third , x cubed , plus , 1 half , x squared , minus x . comma negative 3 less than or equal to x less than or equal to 2 .  A city wants to build a road that goes directly along the x-axis. How many bridges would it have to build?
40. Error Analysis Maurice says: “Every linear function has exactly one zero. It follows from the Fundamental Theorem of Algebra.” Cheryl disagrees. “What about the linear function y = 2 ? y equals 2 question mark ” she asks. “Its graph is a line, but it has no x-intercept.” Whose reasoning is incorrect? Where is the flaw?

Determine whether each of the following statement is always, sometimes, or never true.

41. A polynomial function with real coefficients has real zeros.
42. Polynomial functions with complex coefficients have one complex zero.
43. A polynomial function that does not intercept the x-axis has complex roots only.
44. Reasoning A 4th degree polynomial function has zeros at 3 and 5 − i . 5 minus i .  Can 4 + i 4 plus i  also be a zero of the function? Explain your reasoning.
45. Open-Ended Write a polynomial function that has four possible rational zeros but no actual rational zeros.

46. Three roots of a polynomial equation with real coefficients are 3 , 5 − 3 i , 3 comma 5 minus 3 i comma  and − 3 i . negative 3 i .  Which number MUST also be a root of the equation?

    1. − 3 negative 3
    2. 5 + 3 i 5 plus 3 i
    3. 3 i 3 i
    1. II only
    2. I and II only
    3. II and III only
    4. I, II, and III

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