To analyze the statements regarding the equation \( y = x^4 + 3x^3 - x^2 - 3x \), we can focus on understanding the characteristics of the polynomial.
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Degree of the polynomial: The equation is a polynomial of degree 4. Such a polynomial can have up to 4 real roots (solutions) or a combination of real and non-real roots.
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X-intercepts: The x-intercepts of the graph correspond to the real solutions of the equation. If the graph crosses the x-axis at four distinct points, it means there are four real solutions. If it touches the x-axis at a point without crossing, that indicates a double root or higher multiplicity.
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Turning points: According to calculus, turning points occur where the first derivative equals zero. A degree 4 polynomial can have up to 3 turning points. The number of turning points does not correlate directly with the number of real solutions.
Now, we can review the student statements:
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Statement #1: "The equation has four non-real solutions because there are four x-intercepts."
This statement is false; if there are four x-intercepts, it implies at least four real solutions. Therefore, this statement contradicts itself. -
Statement #2: "The equation has three real solutions because there are three turning points on the graph."
This statement is also false; having three turning points does not necessarily mean there are only three real solutions. A polynomial can have more real solutions than turning points. -
Statement #3: "The equation has four real solutions because there are four x-intercepts."
This statement is true, assuming the graph indeed has four distinct x-intercepts. If the graph crosses the x-axis four times, it indicates there are four real solutions to the equation.
In summary, if the graph indeed has four x-intercepts, then Statement #3 is the correct and true statement.
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