Final answer:
To find the polynomial with zeros 2i, 3, and 0 (with a multiplicity of 3), you multiply the corresponding factors (x - 2i)(x + 2i)(x - 3)x^3 to get P(x) = x^6 - 3x^5 + 4x^4 - 12x^3.
Step-by-step explanation:
To express the product of linear factors with the given zeros of 2i, 3, and 0 with a multiplicity of 3, we must first recognize that each zero corresponds to a factor in the polynomial. For the complex zero 2i, we will have a complex conjugate -2i as well to ensure our polynomial has real coefficients.
The zero at 3 corresponds to a factor of (x - 3), and the zero at 0 with a multiplicity of 3 corresponds to a factor of x^3. Multiplying these factors together gives us the polynomial:
P(x) = (x - 2i)(x + 2i)(x - 3)x^3
Calculating the product of the complex factors yields a quadratic expression, so:
P(x) = (x^2 + 4)(x - 3)x^3
The final step is to multiply all the factors, obtaining:
P(x) = x^6 - 3x^5 + 4x^4 - 12x^3
This polynomial represents the product of the given linear factors and satisfies the zeros provided.