The equation y=x^4 +3x^3 -x^2 -3x has four solutions and is shown in 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.
To determine which of the student statements is true, we need to analyze the given equation and the graph.
First, let's address statement #1: "The equation has four non-real solutions because there are four x-intercepts."
To find the x-intercepts (or solutions) of the equation, we set y equal to zero and solve for x:
0 = x^4 + 3x^3 - x^2 - 3x
However, we cannot determine whether these solutions are real or non-real just based on the fact that there are four x-intercepts. We need to further analyze the equation.
Now, let's move on to statement #2: "The equation has three real solutions because there are three turning points on the graph."
To find the number of turning points on the graph, we look for the number of times the graph changes direction. In this case, the graph is a fourth-degree polynomial, so it can have at most three turning points. However, this doesn't provide enough information to conclude the number of real solutions since a turning point can occur without intersecting the x-axis.
Lastly, statement #3: "The equation has four real solutions because there are four x-intercepts."
As we mentioned earlier, the fact that there are four x-intercepts does not guarantee that all of them are real solutions. To determine if the solutions are real or non-real, we need to further analyze the equation.
In summary, based on the information provided, we cannot determine which of the student statements is true. More analysis of the equation is required to ascertain the nature of the solutions.