To construct a polynomial function of degree 4 with the specified zeros and their multiplicities, let's analyze the given conditions:
- \( x = 3 \) has odd multiplicity. For the sake of degree 4, we can assign it a multiplicity of 1.
- \( x = 2 \) also has odd multiplicity. We can assign it a multiplicity of 1 as well.
- \( x = 5 \) has even multiplicity. To ensure we achieve a total degree of 4, we can assign it a multiplicity of 2.
With these multiplicities, we can write the polynomial in factored form:
\[ f(x) = (x - 3)^1 (x - 2)^1 (x - 5)^2 \]
This can be simplified and expressed as:
\[ f(x) = (x - 3)(x - 2)(x - 5)^2 \]
Now, let's check the degree. The degree contributions are:
- \( x - 3 \) contributes 1 (from multiplicity 1)
- \( x - 2 \) contributes 1 (from multiplicity 1)
- \( (x - 5)^2 \) contributes 2 (from multiplicity 2)
So, the total degree is \( 1 + 1 + 2 = 4 \), which matches our requirement.
Now, looking at the options provided:
- \( f(x) = (x - 3)(x - 2)(x - 5)^2 \) (This is the correct form.)
- \( f(x) = (x - 3)(x - 2)^2(x - 5)^2 \) (Degree is 6, incorrect.)
- \( f(x) = (x - 3)(x - 2)(x - 5)^2 \) (Repeat of the first option.)
- \( f(x) = (x - 3)(x - 2)(x - 5)^2 \) (Repeat of the first option.)
- \( f(x) = (x - 3)^2(x - 2)(x - 5) \) (Degree is 5, incorrect.)
- \( f(x) = (x - 3)^2(x - 2)(x - 5) \) (Repeat of the previous, degree 5.)
- \( f(x) = (x - 3)(x - 2)^2(x - 5) \) (This has degree 5, incorrect.)
- \( f(x) = (x - 3)(x - 2)^2(x - 5) \) (Repeat of the previous, degree 5.)
The correct polynomial function is:
\[ f(x) = (x - 3)(x - 2)(x - 5)^2 \]
Thus the correct response from your selections is:
f(x) = (x - 3)(x - 2)(x - 5)^2