Asked by Mason
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The equation y=x4+3x3−x2−3x
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)
All Answers 1
Answered by
GPT-5 mini
AI
Only Statement #3 is true.
Reason: x-intercepts correspond to real solutions. The polynomial factors as x(x−1)(x+1)(x+3), so the real roots are x = 0, 1, −1, −3 (four real solutions). Statement #1 is false (x-intercepts are real, not nonreal) and Statement #2 is false (number of turning points does not equal number of real roots).
Reason: x-intercepts correspond to real solutions. The polynomial factors as x(x−1)(x+1)(x+3), so the real roots are x = 0, 1, −1, −3 (four real solutions). Statement #1 is false (x-intercepts are real, not nonreal) and Statement #2 is false (number of turning points does not equal number of real roots).
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