SOLUTION: Hi there, I was doing my Advanced functions homework and have been stuck with this question for about an hour now, I would really appreciate if someone could help me with this.

Algebra ->  Graphs -> SOLUTION: Hi there, I was doing my Advanced functions homework and have been stuck with this question for about an hour now, I would really appreciate if someone could help me with this.       Log On


   

Question 190248: Hi there, I was doing my Advanced functions homework and have been stuck with this question for about an hour now, I would really appreciate if someone could help me with this.
Determine an equation of a polynomial function with zeros at x= 2, -2, 1, and y-intercept of 24.
Once again, I could really use the help.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
x=2, x=-2, or x=1 Start with the given solutions


x-2=0, x%2B2=0, or x-1=0 Get every term to the left side


%28x-2%29%28x%2B2%29%28x-1%29=0 Combine the equations to form on equation. (use the zero product property)


a%28x-2%29%28x%2B2%29%28x-1%29=0 Append an "a" (a constant) to the beginning of the left side. This constant will allow us to force the graph to go through the y-intercept of 24.


a%28x%5E2-4%29%28x-1%29=0 FOIL %28x-2%29%28x%2B2%29 to get x%5E2-4 (use the difference of squares)


a%28x%5E3-x%5E2-4x%2B4%29=0 FOIL %28x%5E2-4%29%28x-1%29 to get x%5E3-x%5E2-4x%2B4

So the polynomial with roots of of x=2, x=-2, or x=1 is y=a%28x%5E3-x%5E2-4x%2B4%29


y=a%28x%5E3-x%5E2-4x%2B4%29 Start with the given equation.


24=a%280%5E3-0%5E2-4%280%29%2B4%29 Plug in x=0 and y=24


Note: a y-intercept of 24 means that the point (0,24) is on the graph which means that if x=0, then y=24


24=a%284%29 Multiply and simplify


24%2F4=a Divide both sides by 4.


6=a Divide


So a=6


y=a%28x%5E3-x%5E2-4x%2B4%29 Go back to the previous equation.


y=6%28x%5E3-x%5E2-4x%2B4%29 Plug in a=6


y=6x%5E3-6x%5E2-6%284x%29%2B6%284%29 Distribute


y=6x%5E3-6x%5E2-24x%2B24 Multiply


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Answer:


So y=6x%5E3-6x%5E2-24x%2B24 has the zeros x=2, x=-2, or x=1 and has a y-intercept of 24.