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

3(multiplicity 2), 5+i(multiplicity 1)

since complex roots occur in conjugate pairs,

(x-3)^2 (x-(5+i)) (x-(5-i))

To construct a polynomial function with these given zeros and multiplicities, we will use the fact that complex zeros occur in conjugate pairs. Therefore, if 5+i is a zero, then 5-i must also be a zero with the same multiplicity.

Let's start by writing the factors corresponding to each zero and multiplicity:

For 3 (multiplicity 2), we have (x-3)(x-3).

For 5+i (multiplicity 1) and its conjugate 5-i (multiplicity 1), we have (x - (5+i))(x - (5-i)).

Next, let's multiply these factors out to find the polynomial in factored form:

(x - 3)(x - 3)(x - (5+i))(x - (5-i))

Expanding this expression yields:

(x - 3)(x - 3)(x - 5 - i)(x - 5 + i)

Next, let's simplify this expression:

(x^2 - 6x + 9)(x^2 - 10x + 26)

Now, let's multiply the binomials:

(x^2)(x^2) + (x^2)(-10x) + (x^2)(26) + (-6x)(x^2) + (-6x)(-10x) + (-6x)(26) + (9)(x^2) + (9)(-10x) + (9)(26)

Simplifying, we get:

x^4 - 16x^3 + 80x^2 - 156x + 234

Therefore, the polynomial function of minimum degree with real coefficients whose zeros are 3 (multiplicity 2) and 5+i (multiplicity 1) is:

f(x) = x^4 - 16x^3 + 80x^2 - 156x + 234, in standard form.

To create a polynomial function with the given zeros and their multiplicities, we can start by using the concept of "zeroes" and "multiplicities."

First, we know that the zero 3 has a multiplicity of 2. This means that (x - 3) appears twice as a factor in our polynomial.

Next, we know that the zero 5+i has a multiplicity of 1. This means that (x - (5+i)) appears once as a factor in our polynomial. To match the requirement of having real coefficients, we can assume that the complex conjugate, 5-i, also has multiplicity 1. Therefore, (x - (5-i)) should be another factor.

Now, we can combine all these factors to obtain the polynomial function:

(x - 3)(x - (5+i))(x - (5-i))

To simplify this expression, we can expand the polynomial and express it in standard form:

(x - 3)(x - 5 - i)(x - 5 + i)

Using the distributive property, we can expand the expression:

(x - 3)(x^2 - 5x + xi - 5x + 25 + 5i - xi + 5i - i^2)

Simplifying further:

(x - 3)(x^2 - 10x + 25 + 1)

(x - 3)(x^2 - 10x + 26)

Now, let's multiply the factors:

x(x^2 - 10x + 26) - 3(x^2 - 10x + 26)

x^3 - 10x^2 + 26x - 3x^2 + 30x - 78

Finally, we collect like terms and express the polynomial in standard form:

x^3 - 13x^2 + 56x - 78

Therefore, the polynomial function of minimum degree with real coefficients and the given zeros and multiplicities is:

f(x) = x^3 - 13x^2 + 56x - 78