Final answer:
The polynomial P(x) of degree 5 with zeros 2, 5/2, and 0 (with multiplicity 3) can be constructed by multiplying the factors associated with these zeros to get P(x) = x^3(x - 2)(x - 5/2).
Step-by-step explanation:
The student is asking for the construction of a polynomial P(x) of degree 5 with specified zeros. Since the zeros of a polynomial correspond to the values of x at which the polynomial evaluates to zero, the polynomial can be written using factors associated with these zeros.
The zero 2 contributes a factor of (x - 2). The zero 5/2 contributes a factor of (x - 5/2). The zero 0 with multiplicity 3 contributes a factor of x^3 (because the zero is repeated three times).
Therefore, the polynomial P(x) with the given zeros is:
P(x) = x^3(x - 2)(x - 5/2).
Expanding this and writing it in standard form would give the full polynomial expression.