Question

the equation $y = x^4 + 3x^3 - x^2 - 3x$ 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)

Explanation:

Step1: Analyze Statement #1

x - intercepts are real solutions (where \(y = 0\)). Non - real solutions don't appear on the real - plane graph. So Statement #1 is wrong as x - intercepts mean real solutions.

Step2: Analyze Statement #2

Turning points relate to the degree of the polynomial (for a degree \(n\) polynomial, at most \(n - 1\) turning points) and not directly to the number of real solutions. The number of real solutions is determined by x - intercepts. So Statement #2 is wrong.

Step3: Analyze Statement #3

The graph of \(y=x^{4}+3x^{3}-x^{2}-3x\) has four x - intercepts. Each x - intercept corresponds to a real solution of the equation \(y = 0\) (i.e., \(x^{4}+3x^{3}-x^{2}-3x = 0\)). So the equation has four real solutions because there are four x - intercepts.

Answer:

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