Theorem: If a function y = f(x) has a real root of b, then (x – b) is a factor of f(x).
As given in the problem, there are two roots: –2 and 1/2. The multiplicity of 1/2 is 2 meaning that the root 1/2 repeats twice. So the function f(x) can be written like this.
f(x) = k• (x – (–2))(x – 1/2)^2 = k•(x + 2)(x – 1/2)^2
We're supposed to find the coefficient k to complete the function.
Given that f(–3) = 5, we can plug –3 in for x and 5 in for f(x).
So 5 = k •(–3 + 2)(–3 – 1/2)^2
5 = k(–1)(–7/2)^2
5 = -k•49/4
Then 5 • 4/49 = -k
Or k = –20/49
So the function with the least degree is
f(x) = –20/49 (x + 2)(x – 1/2)^2.
