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anonymous

  • one year ago

Esmeralda and Heinz are working to graph a polynomial function, f(x). Esmeralda says that the third-degree polynomial has four intercepts. Heinz argues that the function only crosses the x-axis three times. Is there a way for them both to be correct? Explain your answer. I really need help! Please :) I am desperate :(

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  1. anonymous
    • one year ago
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    To answer this question, think about how and when a polynomial function crosses the x-axis.

  2. anonymous
    • one year ago
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    I don't know anything about polynomial functions and have to complete this course this week!

  3. cwrw238
    • one year ago
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    well one of the intercepts could be through the y-axis

  4. anonymous
    • one year ago
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    A polynomial function crosses the x-axis at its roots.

  5. anonymous
    • one year ago
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    That is, for example, \[f(x) = ax^3+bx^2+cx+d = 0\]

  6. anonymous
    • one year ago
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    @math1234 so Heinz is correct?

  7. cwrw238
    • one year ago
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    |dw:1435166171785:dw|

  8. anonymous
    • one year ago
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    The roots of a polynomial function are the values of x such that \[f(x) = 0\]

  9. anonymous
    • one year ago
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    Now, notice that a n-order polynomial can have at most n roots at any time.

  10. anonymous
    • one year ago
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    :/ I'm confused

  11. cwrw238
    • one year ago
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    Esmerelda didn't say the intercepts were on the x -axis only . The 3 roots will be at x-intercepts but its possible that the graph cuts they-axis as well

  12. anonymous
    • one year ago
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    Yes, exactly. I wanted to paint that point in the end.

  13. cwrw238
    • one year ago
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    - that way they can both be right

  14. anonymous
    • one year ago
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    So there is a way for them both to be correct because Esmeralda says they can cross four times and heinz just says it crosses the x 3 times

  15. anonymous
    • one year ago
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    First, do you understand that a n-ordered polynomial can have at most n roots, meaning at most n x-axis intercepts?

  16. anonymous
    • one year ago
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    yes.

  17. anonymous
    • one year ago
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    Good, then consider the y-intercept. Polynomials must cross the y axis at some point as they are infinitely continuous.

  18. anonymous
    • one year ago
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    Furthermore, they can only cross the y-axis exactly once.

  19. anonymous
    • one year ago
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    Otherwise, it will violate the straight line test.

  20. cwrw238
    • one year ago
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    - because a 3rd degree polynomial might only cross the x - axis once - it might only have 1 real root The other 2 will be complex .

  21. anonymous
    • one year ago
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    therefore only Esmeralda is right because heinz says it only crosses the x three times. not the y at all right?

  22. anonymous
    • one year ago
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    Not just in the complex case. A 3rd order polynomial can have less x-axis intercepts due to repeated roots.

  23. cwrw238
    • one year ago
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    - yep thats true

  24. anonymous
    • one year ago
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    Okay, so then what is the next step?

  25. anonymous
    • one year ago
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    To conclude, the answer to your question is that both propositions are correct.

  26. cwrw238
    • one year ago
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    No - they are both right because Heinz only refers to the x-axis.

  27. anonymous
    • one year ago
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    okay ? could y'all help with questions related to this?

  28. anonymous
    • one year ago
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    Thank you !! @cwrw238 could you help more?

  29. anonymous
    • one year ago
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    @math1234 could you help more with related question?

  30. cwrw238
    • one year ago
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    I would but gotta go right now sorry

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