SOLUTION: The function p is a fourth-degree polynomial with x-intercepts 1.5, 3, and 8 and y-intercept -3. If p(x) is positive only on the interval (3, 8), find p(x).

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: The function p is a fourth-degree polynomial with x-intercepts 1.5, 3, and 8 and y-intercept -3. If p(x) is positive only on the interval (3, 8), find p(x).      Log On


   

Question 850059: The function p is a fourth-degree polynomial with x-intercepts 1.5, 3, and 8 and y-intercept -3. If p(x) is positive only on the interval (3, 8), find p(x).
Answer by josgarithmetic(39621) About Me  (Show Source):
You can put this solution on YOUR website!
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%28x%29=-%28x-1.5%29%5E2%28x-3%29%28x-8%29.

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%280%29=-3=%28x-1.5%29%5E2%28x-3%29%28x-8%29%28x-d%29, and solved for d; but the resulting d=-%281%2F12%29 did not work for the interval requirement. Neither did the opposite, y=-%28x-1.5%29%5E2%28x-3%29%28x-8%29%28x%2B1%2F12%29.

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.