Question

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

To evaluate the statements regarding the polynomial 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. The presence of x-intercepts indicates that there are real solutions. Therefore, if there are four x-intercepts, the solutions are real, not non-real.

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 (local maxima and minima) in a polynomial graph is typically less than or equal to the degree of the polynomial minus one. A quartic polynomial such as this one may have up to three turning points but can have four real roots (solutions).

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

   • This statement is true. If the graph crosses the x-axis at four distinct points, it means that the polynomial has four real solutions corresponding to those x-intercepts.

Based on this reasoning, Statement #3 is true.