To determine which statements about the equation \( y = x^4 + 3x^3 - x^2 - 3x \) are true, we can analyze the polynomial and its graph.
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Degree and number of solutions: The polynomial is of degree 4, which generally means it can have up to 4 real roots. Since it states that there are four solutions, it's likely that all roots of the polynomial are real.
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Behavior of the polynomial: As \( x \to \pm \infty \), \( y = x^4 + 3x^3 - x^2 - 3x \) will tend to \( +\infty \) because the leading term \( x^4 \) dominates. Therefore, the ends of the graph will rise towards positive infinity on both sides.
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Critical points: To identify how many times the curve crosses the x-axis (i.e., solutions to \( y = 0 \)), you would typically find the critical points of the function by setting the derivative \( y' \) to zero and analyzing the sign changes (which reflects the behavior of the function).
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Nature of roots: If the polynomial has four real solutions, these could be distinct or could have multiplicity (i.e., some roots could be repeated). Additionally, the curve may touch the x-axis without crossing it at certain points if any real roots have an even multiplicity.
To analyze the given statements, you would need to evaluate specifics such as whether the roots are distinct, whether any are complex, or whether there are conditions relating to the derivative, etc.
If you provide the list of specific statements you have in mind, I can help to judge which one is true based on our analysis.