Answer:
f(x) = x^4 + (5/2)x^3 + x^2 + (7/2)x + (5/2)
Explanation:
To find a polynomial function with the given zeros and f(1) = 18, we can start by using the zero-product property to set up the factors of the polynomial.
Given zeros:
-2, -1/2, i
Since i is a complex number, its conjugate, -i, is also a root of the polynomial.
The factors of the polynomial can be written as:
(x + 2)(x + 1/2)(x - i)(x + i)
To find the polynomial, we can multiply these factors together:
(x + 2)(x + 1/2)(x - i)(x + i) =
(x^2 + 2x + (1/2)x + 1)(x^2 + 1) =
(x^2 + (5/2)x + 1)(x^2 + 1) =
x^4 + (5/2)x^3 + x^2 + (5/2)x + x + (5/2) =
x^4 + (5/2)x^3 + x^2 + (7/2)x + (5/2)
Now we have the polynomial function in standard form:
f(x) = x^4 + (5/2)x^3 + x^2 + (7/2)x + (5/2)
To find f(1), we substitute x = 1 into the polynomial:
f(1) = (1)^4 + (5/2)(1)^3 + (1)^2 + (7/2)(1) + (5/2)
f(1) = 1 + (5/2) + 1 + (7/2) + (5/2)
f(1) = 18
Therefore, the polynomial function that satisfies the given conditions is:
f(x) = x^4 + (5/2)x^3 + x^2 + (7/2)x + (5/2)