Question
A polynomial function contains the factors
,
, and
. Which graph(s) below could represent the graph of this function?
A
I, only
B
II, only
C
I and III
D
I, II, and III
Answers
Answered by
GPT-4o mini
To determine which graph(s) could represent the polynomial function containing certain factors, we need to consider the key features of the graph based on the factors provided.
1. **Factors:** The polynomial's factors will influence the x-intercepts of the graph. If a factor is of the form (x - r), then the graph will cross the x-axis at x = r. If a factor is of the form (x - r)^n, where n is an even integer, the graph will touch the x-axis at x = r but not cross it. If n is an odd integer, the graph will cross the x-axis.
2. **Multiplicity:** If the graph has a zero of odd multiplicity, it will cross the x-axis at that zero. If it has a zero of even multiplicity, it will touch the x-axis and turn back.
You need to examine the graphs and see if they show the appropriate x-intercepts and behavior at those intercepts based on the factors mentioned.
Without the specific factors or descriptions of the graphs provided, I can't give you a direct answer. However, based on this method, you can analyze each graph in question.
- Look for the x-intercepts corresponding to the factors.
- Check if the graph crosses or touches the axis at those intercepts, depending on whether the factors have odd or even multiplicities.
Once you assess each of the given graphs based on these rules, you can conclude which options are valid. Depending on your observations, you can choose from A (I only), B (II only), C (I and III), or D (I, II, and III).
1. **Factors:** The polynomial's factors will influence the x-intercepts of the graph. If a factor is of the form (x - r), then the graph will cross the x-axis at x = r. If a factor is of the form (x - r)^n, where n is an even integer, the graph will touch the x-axis at x = r but not cross it. If n is an odd integer, the graph will cross the x-axis.
2. **Multiplicity:** If the graph has a zero of odd multiplicity, it will cross the x-axis at that zero. If it has a zero of even multiplicity, it will touch the x-axis and turn back.
You need to examine the graphs and see if they show the appropriate x-intercepts and behavior at those intercepts based on the factors mentioned.
Without the specific factors or descriptions of the graphs provided, I can't give you a direct answer. However, based on this method, you can analyze each graph in question.
- Look for the x-intercepts corresponding to the factors.
- Check if the graph crosses or touches the axis at those intercepts, depending on whether the factors have odd or even multiplicities.
Once you assess each of the given graphs based on these rules, you can conclude which options are valid. Depending on your observations, you can choose from A (I only), B (II only), C (I and III), or D (I, II, and III).