Write a polynomial f(x) that satisfies the given conditions.
Polynomial of lowest degree with zeros of -4 (multiplicity 1), 3 (multiplicity 3), and with f (0) = 216.
To find a polynomial that satisfies the given conditions, we first determine the factors of the polynomial based on the zeros and their multiplicities.
The zero -4 has a multiplicity of 1, so its factor is (x + 4).
The zero 3 has a multiplicity of 3, so its factor is (x - 3)^3.
Therefore, the polynomial is:
f(x) = a(x + 4)(x - 3)^3.
Now we can use the given point (0, 216) to solve for the value of a:
f(0) = a(0 + 4)(0 - 3)^3 = a(4)(-27) = 216,
-108a = 216,
a = -2.
Thus, the polynomial that satisfies the given conditions is:
f(x) = -2(x + 4)(x - 3)^3.