SOLUTION: The function f is a fifth-degree polynomial help? The function f is a fifth-degree polynomial with the x-intercepts -1, 4, and 9, y-intercept 36 and f (x) ≥ 0 for x &#

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Question 911773: The function f is a fifth-degree polynomial help?
The function f is a fifth-degree polynomial with
the x-intercepts -1, 4, and 9,
y-intercept 36 and
f (x) ≥ 0 for x ≤ 9.
Find f (x).
f (x) =
Please explain,
Thanks
Answer by rothauserc(4718)   (Show Source): You can put this solution on YOUR website!
We are given; the x-intercepts -1, 4, and 9,
y-intercept 36 and
f(x) ≥ 0 for x ≤ 9, therefore
f(x) = (x+1)*(x-4)*(x-9)*g(x) where
g(x) is a quadratic equation of the form
Ax^2 + Bx +C
To determine g(x):
We need f(x) ≥ 0 for x ≤ 9.
==> (x + 1)(x - 4)(x - 9)g(x) ≥ 0 for x ≤ 9.
Since we need g(x) to be quadratic, we can take g(x) = A(x + 1)(x - 4) for some constant A and
Now, we have f(x) = A(x + 1)^2*(x - 4)^2*(x - 9) for some A.
we use f(0) = A(1)^2*(-4)^2*(-9) = 36
36 = A*(1)*16*(-9)
36 = -A*144
A = -1/4
therefore
f(x) = (-1/4)*(x + 1)^2*(x - 4)^2*(x - 9)
f(x) = -x^5/4+(15*x^4)/4-(55*x^3)/4-(15*x^2)/4+50*x+36

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