Question 959245: Having a lot of difficulty visualizing this don't know where to put the exponents could really use explanations:
The function f is a fifth-degree polynomial with
the x-intercepts -5, 2, and 10,
y-intercept 500 and
f (x) ≥ 0 for x ≤ 10.
Find f (x).
f (x) =
I don't understand what f(x)≥0 for x≤10 means, please explain that and how to solve for the rest.
Thank you
Answer by MathLover1(20849)   (Show Source):
You can put this solution on YOUR website! given:
a fifth-degree polynomial with
the x-intercepts  ,  , and  ,
y-intercept  and
 for 
By hypothesis,  has factors  ,  , and by the Factor Theorem.
==>  for some 2nd degree polynomial  .
To determine :
We need  for  .
==>  for
Note that each factor is negative for some .
There a few ways to deal with this; here is one way.
Note that both  and  change signs when , while  does not.
Since we need to be quadratic, we can take  for some constant  .
Now, we have  for some .
Note: that we have not changed the x-intercepts by repeating some of the factors
Since  is never negative, needing  for  reduces to needing  for  .
*This is guaranteed if  .
Finally, we use  to determine :
==> , which is indeed negative.
Hence, we can take
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