Write a polynomial function of minimum degree in standard form with real coefficients whose zeros and their multiplicities include those listed.

2(multiplicity 2), -4(multiplicity 3)

y = (x-2)^2 (x+4)^3

now just expand that.

Sure, let's come up with a polynomial function that perfectly matches your criteria. Since we know the zeros and their multiplicities, we can write the function as follows:

f(x) = (x - 2)²(x + 4)³

Here, we have the zero 2 with multiplicity 2, represented by (x - 2)², and the zero -4 with multiplicity 3, represented by (x + 4)³. By multiplying these factors together, we obtain a polynomial function that meets your requirements.

To form a polynomial function with the given zeros and multiplicities, we need to include each zero with its corresponding multiplicity.

Let's start by considering the zero 2 with a multiplicity of 2. Since the multiplicity is 2, it means that the factor (x - 2) will occur twice in the polynomial.

Next, we move on to the zero -4 with a multiplicity of 3. Similarly, since the multiplicity is 3, the factor (x + 4) will be present three times.

Now, we can write the polynomial function as follows:

f(x) = (x - 2)(x - 2)(x + 4)(x + 4)(x + 4)

Multiplying this out, we get:

f(x) = (x - 2)^2(x + 4)^3

Expanding this further, we have:

f(x) = (x - 2)(x - 2)(x + 4)(x + 4)(x + 4)
= (x^2 - 4x + 4)(x^3 + 12x^2 + 48x + 64)
= x^5 + 8x^4 + 20x^3 + 8x^2 - 384x - 512

Thus, the polynomial function of minimum degree with real coefficients, whose zeros and multiplicities include 2 (multiplicity 2) and -4 (multiplicity 3), is f(x) = x^5 + 8x^4 + 20x^3 + 8x^2 - 384x - 512.

To write a polynomial function with the given zeros and their multiplicities, we can use the fact that zeros of a polynomial correspond to the factors of the polynomial.

The given zeros are 2 with a multiplicity of 2 and -4 with a multiplicity of 3. This means that (x - 2) and (x + 4) are factors of the polynomial.

To find the polynomial, you need to perform the following steps:

Step 1: Start with each zero and its multiplicity:
For the zero 2 with multiplicity 2, we have (x - 2)^2 as a factor.
For the zero -4 with multiplicity 3, we have (x + 4)^3 as a factor.

Step 2: Multiply the factors together:
Multiply the factors obtained in step 1 to get the final polynomial function.

Therefore, the polynomial function of minimum degree in standard form with the given zeros and their multiplicities is:
f(x) = (x - 2)^2 * (x + 4)^3

You can expand this polynomial to get the complete expression if needed.