Use the image to answer the question.

Question

Use the image to answer the question.

A curve is graphed on a coordinate plane. The plane’s x-axis ranges from negative 5 to 5 in unit increments but labeled every 5 units. The y-axis ranges from negative 10 to 10 in unit increments but labeled at every 5 units.

The equation y=x4+3x3−x2−3x
has four solutions and is shown in the graph. Which of the following student statements is true?

Statement #1: The equation has four non-real solutions because there are four x-intercepts.

Statement #2: The equation has three real solutions because there are three turning points on the graph.

Statement #3: The equation has four real solutions because there are four x-intercepts.

(1 point)
Statement #
is true.

Answer 1

Bot GPT 4-o mini answered
13 hours ago

To determine which statement is true, we need to analyze the equation and its graphical representation.

Analyzing the Statements

Statement #1: "The equation has four non-real solutions because there are four x-intercepts."
- This statement is false. If there are four x-intercepts, it means the equation has four real solutions since x-intercepts correspond to the solutions of the equation \( y = 0 \).
Statement #2: "The equation has three real solutions because there are three turning points on the graph."
- This statement is also false. The number of turning points does not directly correlate with the number of solutions. A polynomial can have more turning points than solutions.
Statement #3: "The equation has four real solutions because there are four x-intercepts."
- This statement is true. The equation being a polynomial of degree 4 can have up to 4 real solutions, and having four x-intercepts indicates that there are indeed four real solutions.

Conclusion

Thus, the correct statement is:

Statement #3 is true.