Question

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Question
Write a polynomial function of degree 4 with the following characteristics:

Degree: 4

Zeros: x = 3
(odd multiplicity) x = 2
(odd multiplicity) x = 5
(even multiplicity)



Which of the following is the correct polynomial?

(3 points)
Responses

f(x) = (x−3)(x−2)2(x−5)2
f(x) = (x−3)(x−2)2(x−5)2

f(x) = (x−3)(x−2)(x−5)2
f(x) = (x−3)(x−2)(x−5)2

f(x) = (x−3)2(x−2)(x−5)
f(x) = (x−3)2(x−2)(x−5)

f(x) = (x−3)(x−2)2(x−5)

Answer 1

To construct a polynomial function of degree 4 with the given characteristics about its zeros, we need to account for their multiplicities:

- The zero \( x = 3 \) has an odd multiplicity (which could be 1 or 3).
- The zero \( x = 2 \) has an odd multiplicity (which could be 1 or 3).
- The zero \( x = 5 \) has an even multiplicity (which could be 2 or 4).

To ensure the total degree is 4, we can establish that:

1. Let’s assume \( x = 3 \) has multiplicity 1.
2. Let’s assume \( x = 2 \) has multiplicity 1.
3. Consequently, \( x = 5 \) must have multiplicity 2 to reach a total degree of 4.

Thus, the polynomial can be expressed as:

\[
f(x) = (x - 3)(x - 2)(x - 5)^2
\]

Now, let's consider the options provided in your question:

1. **f(x) = (x−3)(x−2)²(x−5)²** — Degree 6 (Not correct).
2. **f(x) = (x−3)(x−2)(x−5)²** — Degree 4 (This matches our polynomial).
3. **f(x) = (x−3)²(x−2)(x−5)** — Degree 4 (This does not match multiplicity conditions).
4. **f(x) = (x−3)(x−2)²(x−5)** — Degree 5 (Not correct).

Among the options, the correct polynomial function that fulfills the criteria is:

\[
f(x) = (x−3)(x−2)(x−5)²
\]

This polynomial has a degree of 4, with the correct multiplicities corresponding to the specified zeros.