Use the given graph of the polynomial function to complete the following.

Question

a. Estimate the​ x-intercept(s).
b. State whether the leading coefficient is positive or negative.
c. Determine whether the polynomial function is cubic or quartic.
-5
-3
-1
1
3
5
-5
-3
-1
1
3
5
x
y

A coordinate system has a horizontal x-axis labeled from negative 5 to 5 in increments of 1 and a vertical y-axis labeled from negative 5 to 5 in increments of 1. From left to right, a curve rises steeply at a decreasing rate passing through the point (negative 2, 0) to a local maximum, falls at an increasing then decreasing rate passing through the points (0, 1.2) and left parenthesis 1 comma 0 right parenthesis to a local minimum comma and rises steeply at an increasing rate passing through the point left parenthesis 3 comma 0 right parenthesis.
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Question content area right
Part 1
a. The​ x-intercepts are xequals
  
enter your response here.
​(Use a comma to separate answers as​ needed.)
Part 2
b. What is the sign of the leading​ coefficient?
Negative
Positive
Part 3
c. What is the nature of the polynomial whose graph is shown in the​ figure?

Answer 1

To analyze the given polynomial function and complete the tasks:

Part 1: Estimate the x-intercept(s)

From your description of the graph, the x-intercepts seem to be located at:

• \( x = -2 \)
• \( x = 1 \)
• \( x = 3 \)

Thus, the x-intercepts are:

• x = -2, 1, 3

Part 2: State whether the leading coefficient is positive or negative

To determine the sign of the leading coefficient, we can observe the behavior of the polynomial as \( x \) approaches positive and negative infinity:

• You mentioned that the curve "rises steeply at a decreasing rate" on the left side, indicating that as \( x \) approaches negative infinity, \( y \) also approaches positive infinity.
• Conversely, as \( x \) approaches positive infinity, the curve rises steeply which indicates that \( y \) also approaches positive infinity.

Since the graph rises to the right and left, the leading coefficient of the polynomial is Positive.

Part 3: Determine whether the polynomial function is cubic or quartic

The polynomial has three x-intercepts (which typically indicates it could be a cubic polynomial), and it appears to change direction (local maximum/minimum) at several points. However, the presence of three x-intercepts alone is not enough to determine if the polynomial is cubic or quartic without additional local behaviors.

Typically, a cubic polynomial can have up to 2 turning points, whereas a quartic polynomial can have up to 3. Given the description of having more than one local maximum and minimum, this suggests the polynomial likely has four total real roots.

Based on the information provided:

• The polynomial is likely quartic.

Summary of Answers

a. The x-intercepts are: -2, 1, 3
b. The leading coefficient is: Positive
c. The nature of the polynomial is: Quartic