Question 1154932: Which of the formulas below could be a polynomial with all of the following properties: its only zeros are x=−9,−3,3, it has its y-intercept at y=4, and its long-run behavior is y→−∞ as x→±∞? Select every formula that has all of these properties.
A. y=−4/729(x−9)(x−3)(x+3)
B. y=−4/81(x+9)(x+3)(x−3)
C. y=−4/59049(x+9)4(x+3)(x−3)
D. y=−4/243(x+9)(x+3)(x−3)2
E. y=−4/729(x+9)2(x+3)(x−3)
F. y=−4/243(x+9)(x+3)2(x−3)
G. y=4/243(x+9)(x+3)(x−3)2
Answer by greenestamps(13200)   (Show Source):
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A. 
B. 
C. 
D. 
E. 
F. 
G. 
The specified long-term behavior means the polynomial has to be even degree with a negative leading coefficient. Choices C, D, E, and F satisfy that requirement.
With zeros at -9, -3, and 3, the binomial factors have to be (x+9), (x+3), and (x-3). Choices C, D, E, and F satisfy that requirement also.
Any of those choices that have a y-intercept of 4 will satisfy all the requirements.
Without doing the multiplication, it can be seen that the y-intercept for choice D is negative, while the y-intercepts for choices C, E, and F will be positive.
Any of those three will satisfy all the requirements if the y-intercept is 4.
Choice C: 
Choice E: 
Choice F: 
ANSWER: C, E, and F are all polynomials that satisfy all of the given conditions.
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