We know that a polynomial of degree 3 has three roots. Since p(x) has a root of multiplicity 2 at x=2, we can write one factor of the polynomial as (x-2)^2. Similarly, since p(x) has a root of multiplicity 1 at x=-4, we can write another factor as (x+4). Thus, the polynomial can be written as:
p(x) = k(x-2)^2(x+4)
where k is some constant.
To find the value of k, we use the fact that the y-intercept is -3.2, which means that p(0) = -3.2. Substituting x=0 into the equation for p(x), we get:
p(0) = k(-2)^2(4) = 16k
Setting this equal to -3.2, we have:
16k = -3.2
k = -0.2
So the polynomial is:
p(x) = -0.2(x-2)^2(x+4)
Therefore, the polynomial of degree 3, p(x), with a root of multiplicity 2 at x=2 and root of multiplicity 1 at x=-4, and y-intercept of -3.2, is given by:
p(x) = -0.2(x-2)^2(x+4)