Write a polynomial f (x) that meets the given conditions.Polynomial of lowest degree with zeros of -2 (multiplicity 2) and 3 (multiplicity 2) and with f(0)=−108𝑓(0)=−108Select one:a. f (x) = -3x4 + 6x3 + 33x2 - 18x - 108b. f (x) = -3x4 + 6x3 + 33x2 - 36x - 108c. f (x) = -3x4 + 6x3 - 111x2 - 36x - 108d. f (x) = x4 - 2x3 - 11x2 + 12x - 108
Question
Write a polynomial f (x) that meets the given conditions.Polynomial of lowest degree with zeros of -2 (multiplicity 2) and 3 (multiplicity 2) and with f(0)=−108𝑓(0)=−108Select one:a. f (x) = -3x4 + 6x3 + 33x2 - 18x - 108b. f (x) = -3x4 + 6x3 + 33x2 - 36x - 108c. f (x) = -3x4 + 6x3 - 111x2 - 36x - 108d. f (x) = x4 - 2x3 - 11x2 + 12x - 108
Solution
The polynomial of lowest degree with zeros of -2 (multiplicity 2) and 3 (multiplicity 2) can be written as:
f(x) = a*(x+2)^2*(x-3)^2
We know that f(0) = -108. Substituting x = 0 in the equation, we get:
-108 = a*(-2)^2*(3)^2 = 36a
Solving for a, we get a = -108/36 = -3.
So, the polynomial is:
f(x) = -3*(x+2)^2*(x-3)^2
Expanding this, we get:
f(x) = -3x^4 + 6x^3 + 33x^2 - 36x - 108
So, the correct answer is b. f (x) = -3x^4 + 6x^3 + 33x^2 - 36x - 108.
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