Question

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

Degree: 4

Zeros: x = 3
𝑥

=

3
(odd multiplicity) x = 2
𝑥

=

2
(odd multiplicity) x = 5
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)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)(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)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 zeros and their respective multiplicities, we can follow these guidelines:

1. **Degree**: The polynomial must be of degree 4.
2. **Zeros**:
- \( x = 3 \) with an **odd multiplicity** (let's use multiplicity 1).
- \( x = 2 \) with an **odd multiplicity** (let's use multiplicity 1).
- \( x = 5 \) with an **even multiplicity** (let's use multiplicity 2).

Considering these conditions, the corresponding factors of the polynomial are:
- For \( x = 3 \): \( (x - 3) \)
- For \( x = 2 \): \( (x - 2) \)
- For \( x = 5 \): \( (x - 5)^2 \)

Therefore, we can combine these to form the polynomial:
\[
f(x) = (x - 3)(x - 2)(x - 5)^2
\]

Next, we verify the characteristics:
- Degree is \( 1 + 1 + 2 = 4 \) (correct).
- Zeros and multiplicities:
- \( x = 3 \): multiplicity 1 (odd).
- \( x = 2 \): multiplicity 1 (odd).
- \( x = 5 \): multiplicity 2 (even).

Thus, the correct polynomial function is:
\[
f(x) = (x - 3)(x - 2)(x - 5)^2
\]

The corresponding option in your provided choices is:
- \( f(x) = (x - 3)(x - 2)(x - 5)^2 \)

This option is the correct polynomial.