Construct a polynomial function with the following properties: third degree, 2 is a zero of multiplicity 2, -2 is the only other zero, leading coefficient is 4.

Step-by-step explanation:

p(x) = 4 (x - 2)^2 (x - (-2)), since, in general,  a is a zero of multiplicity m of a polynomial if (x - a)^m is a factor of the polynomial and (x-a)^(m+1) is not a factor, and A is the leading coefficient of the polynomial if the polynomial is written as A(x - a1)^m1 (x - a2)^m2 ... (x - aN)^mN.

Thus,

p(x) = 4 (x - 2)^2 (x + 2)

p(x) = 4 (x - 2)(x^2 - 4)

p(x) = 4 ( x^3 -4x - 2x^2 + 8)

p(x) = 4x^3 - 16x - 8x^2 + 32

p(x) = 4x^3 - 8x^2 - 16x + 32