To analyze the statements regarding the equation \( y = x^4 + 3x^3 - x^2 - 3x \), let's consider the definition of real and non-real solutions and the implications of x-intercepts and turning points.
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x-intercepts: These occur when \( y = 0 \). If the graph of the function crosses the x-axis at certain points, those are the real solutions of the equation.
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Turning points: These are points where the derivative of the function changes sign, which include local maxima, minima, or points of inflection. The number of turning points can give an indication of the behavior of the function but does not necessarily correlate directly with the number of real solutions.
Now, let's analyze the statements based on the definitions:
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Statement #1: "The equation has four non-real solutions because there are four x-intercepts."
- This statement is incorrect. If there are four x-intercepts, that indicates there are four real solutions, not non-real.
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Statement #2: "The equation has three real solutions because there are three turning points on the graph."
- This statement is also incorrect. The number of turning points does not directly correspond to the number of solutions; it is possible to have more solutions than turning points or vice versa.
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Statement #3: "The equation has four real solutions because there are four x-intercepts."
- This statement is likely to be true. If the graph of the polynomial intersects the x-axis at four distinct points, this indicates that there are indeed four real solutions.
Based on the analysis above, the true statement is Statement #3: "The equation has four real solutions because there are four x-intercepts."