Question

Consider the graph of the polynomial function p(x). Given that (2x+3)
is a factor of p(x), which of the following could be the multiplicity of (2x+3)
?

(1 point)
Responses

2
2

1.5
1.5

−1.5
negative 1.5

1

Answer 1

The polynomial function \( p(x) \) has a factor \( (2x + 3) \). According to the properties of polynomial factors, the multiplicity of a factor must be a non-negative integer. This means that it can only take values such as 0, 1, 2, 3, etc.

Now, let's analyze the provided options:

- \( 2 \): This is a valid multiplicity (even).
- \( 1.5 \): This is not a valid multiplicity since it is not an integer.
- \( -1.5 \): This is also not a valid multiplicity since it is negative.
- \( 1 \): This is a valid multiplicity (odd).

Thus, the valid multiplicities for the factor \( (2x + 3) \) are \( 2 \) and \( 1 \). Therefore, the potential multiplicities of the factor \( (2x + 3) \) could be either \( 2 \) or \( 1 \).

From the given options, the valid multiplicities are:
- 2
- 1

So, you can choose either \( 2 \) or \( 1 \) as potential multiplicities of \( (2x + 3) \).