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

Question

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

To evaluate the statements provided in relation to the equation \( y = x^4 + 3x^3 - x^2 - 3x \):

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

   • This statement is false. If an equation has x-intercepts, it signifies that there are real solutions at those intercepts.

2. 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 equate to the number of solutions since turning points can occur at real solutions but may not account for all solutions. In fact, a polynomial can have more solutions than turning points.

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

   • This statement is true. A polynomial equation can have as many real solutions as it has x-intercepts. If the graph of the polynomial shows four x-intercepts, it indicates that there are four real solutions.

Thus, the true statement is Statement #3.