Final answer:
The equation of the polynomial with given roots and y-intercept is derived by constructing it from its factored form and solving for the leading coefficient using the given y-intercept. The final polynomial is expressed as p(x) = (1/2)(x - 4)^2(x + 3)(x - 1).
Step-by-step explanation:
To find the equation of a polynomial given specific roots and a y-intercept, we need to construct it based on the given information. A root of multiplicity 2 at 4 means the factor associated with that root is (x - 4)^2. Roots of multiplicity 1 at -3 and 1 mean the factors are (x + 3) and (x - 1), respectively. So the polynomial in its factored form is p(x) = a(x - 4)^2(x + 3)(x - 1), where a is the leading coefficient that we need to determine.
Since the polynomial has a y-intercept at (0,24), we can substitute x = 0 in the polynomial and set p(0) = 24 to find the value of a. This gives us 24 = a(-4)^2(3)(-1), which simplifies to 24 = 48a. Dividing both sides by 48 gives a = 1/2. Our polynomial equation is therefore p(x) = \(\frac{1}{2}\)(x - 4)^2(x + 3)(x - 1).
Expanding this out to get the standard polynomial form, we would perform the multiplication of the factors, keeping in mind the leading coefficient of \(\frac{1}{2}\).