Below is a graph of a cubic (3rd degree) polynomial. A student was asked to explain what he knew about the solutions. He said that the graph had 3 non-real complex number solutions. Is the student correct? Why or why not? How do you know? If he's not correct, what is the correct way to describe the solutions? (4 points)

Answer 1

To determine whether the student's statement about the cubic polynomial is correct, we need to analyze the characteristics of a cubic function and its graph.

1. Degree of the Polynomial: A cubic polynomial has a degree of 3, which means it can have up to three solutions (roots) in total. These can be real or complex.

2. Types of Solutions: The nature of the solutions (real vs. non-real complex) is determined by the behavior of the polynomial graph:

   • If the graph intersects the x-axis three times, there are three real solutions.
   • If the graph intersects the x-axis once and has a "double" root (where it touches the x-axis but does not cross it), there are two real solutions and one real solution.
   • If the graph does not intersect the x-axis at all (meaning it stays entirely above or below the x-axis), there are three non-real complex solutions.

3. Verification Against the Graph: Since we don't have the actual graph to reference, we can use the above points to explain how to analyze it. You can observe if the graph crosses the x-axis:

   • If it crosses at three distinct points, the student is incorrect; there are three real solutions.
   • If it crosses at one distinct point and touches at another, then there are two real solutions and one double root.
   • If it does not cross the x-axis at all, then the student's statement would be correct, as there would be three non-real complex solutions.

4. Conclusion: The student’s assertion that the graph of a cubic polynomial has "3 non-real complex solutions" can only be correct if the graph indeed does not intersect the x-axis at all. If it intersects the x-axis at any point, the number of real solutions will be at least one, making the student's claim incorrect.

Correct Description of Solutions: If the graph shows any intersection with the x-axis, then describe the solutions according to the number of intersections (real solutions). If the graph does not intersect, then the correct description would be “the polynomial has three non-real complex solutions.”

To definitively assess the student's correctness, one would need to visually examine the graph in question.