Question 850059
REMOVED.
I had solved this but now disagree with my solution.


I would really need to fully re-solve this, but the actual equation in factored form will be {{{p(x)=-(x-1.5)^2(x-3)(x-8)}}}.


Second solution, unrefined, was that either one of the factors were repeated or that a new unknown factor x-d would be needed.  This was because degree four polynomial function must have four binomial factors, or in some way have a x^4 term when in general form.


I had tried {{{p(0)=-3=(x-1.5)^2(x-3)(x-8)(x-d)}}}, and solved for d; but the resulting {{{d=-(1/12)}}} did not work for the interval requirement.  Neither did the opposite, {{{y=-(x-1.5)^2(x-3)(x-8)(x+1/12)}}}.


Testing for a repeated binomial factor, found was exactly one interval over which the function were above or below the x-axis while all the other intervals were the opposite.  I then picked the sign necessary to let the y-intercept be -3.  The function shown at the top of this solution post was the one that worked.