SOLUTION: The function p is a fourth-degree polynomial with x-intercepts 1, 4, and 10 and y-intercept -2. If p(x) is positive only on the interval (4, 10), find p(x).
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Question 916690: The function p is a fourth-degree polynomial with x-intercepts 1, 4, and 10 and y-intercept -2. If p(x) is positive only on the interval (4, 10), find p(x).
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
A 4th degree polynomial must have 4 zeroes. If any of them are complex numbers, then an even number of them must be complex numbers. But we are given three real zeroes; hence the 4th zero must be real and therefore one of the zeros 1, 4, or 10 must have a multiplicity of 2. Since the polynomial is only positive on (4,10), the graph must cross (not be tangent to) the x-axis at 4 and 10. That leaves 1 as the zero that must have the multiplicity of 2. Therefore:
\ =\ a(x\ -\ 1)^2(x\ -\ 4)(x\ -\ 10))
Multiply this out, then choose 
such that \ =\ -2)
. Hint: Choose such that the constant term equals -2.
John
My calculator said it, I believe it, that settles it
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