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)

Answer 1

You cant solve with just the numbers

Answer 2

The factor \( (2x + 3) \) corresponds to a root at \( x = -\frac{3}{2} \). The multiplicity of this factor can be any positive integer (1, 2, 3, etc.).

If a factor has an **odd multiplicity**, the graph of the polynomial will cross the x-axis at that root. If the factor has an **even multiplicity**, the graph will touch the x-axis at that root but will not cross it.

Without additional information about how the graph behaves at the root or other factors of the polynomial, any positive integer (1, 2, 3, ...) could be a possible multiplicity for the factor \( (2x + 3) \).

If you are provided with specific options, please share them for a more focused answer. Otherwise, the multiplicity could be any positive integer.

Answer 3

You're correct; without specific options to choose from, it’s challenging to narrow down a single answer regarding the potential multiplicity of the factor \( (2x + 3) \).

Here's a general guideline regarding possible multiplicities for the factor \( (2x + 3) \):

1. **Odd Multiplicity (e.g., 1, 3, 5,...)**:
- The graph crosses the x-axis at \( x = -\frac{3}{2} \).

2. **Even Multiplicity (e.g., 2, 4, 6,...)**:
- The graph touches the x-axis at \( x = -\frac{3}{2} \) but does not cross it.

In conclusion, the multiplicity of the factor \( (2x + 3) \) could be any positive integer (1, 2, 3, 4, etc.). If you have specific options or additional details about the behavior of the polynomial graph, I can provide a more targeted answer based on that information.