Question 953541: The function p is a fourth-degree polynomial with x-intercepts 0.5, 5, and 10 and y-intercept -1. If p(x) is positive only on the interval (5, 10), find p(x).
How would I go about solving?
Thank you
Found 2 solutions by stanbon, MathLover1:
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! The function p is a fourth-degree polynomial with x-intercepts 0.5, 5, and 10 and y-intercept -1. If p(x) is positive only on the interval (5, 10), find p(x).
How would I go about solving?
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Given those x-intercepts you get::
y = a(2x-1)(x-5)
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With the y-intercept you can solve for "a":
10 = a(-1)(-5)
a = 2
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Equation:
y = 2(2x-1)(x-5)
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Note:: Being a quadratic it cannot be positive on (5,10) if it has
x intercepts 0.5 and 5.
Cheers,
Stan H.
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Answer by MathLover1(20850)  (Show Source):
You can put this solution on YOUR website! The function  is a fourth-degree polynomial with x-intercepts  ,  , and  and y-intercept  .
If  is positive only on the interval (, ), find .
A fourth-degree polynomial must have  roots.  are given. Presumably  of those three has multiplicity of  , as opposed to there being an entirely separate root that the problem isn't telling about.
and can't either have a multiplicity of without violating the condition about "positive only on the interval (, )", so that leaves or  to be the multiple root.
Use the factor theorem to convert the four roots into factors:
Multiply these factors together:
The y-intercept of this function is  , so to achieve the desired y-intercept, multiply all the coefficients by  :
as you can see on a graph,
x-intercepts are , , and
y-intercept is ,
and
is positive only on the interval (, )
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