anonymous
  • anonymous
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 :(
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
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anonymous
  • anonymous
To answer this question, think about how and when a polynomial function crosses the x-axis.
anonymous
  • anonymous
I don't know anything about polynomial functions and have to complete this course this week!
cwrw238
  • cwrw238
well one of the intercepts could be through the y-axis

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More answers

anonymous
  • anonymous
A polynomial function crosses the x-axis at its roots.
anonymous
  • anonymous
That is, for example, \[f(x) = ax^3+bx^2+cx+d = 0\]
anonymous
  • anonymous
@math1234 so Heinz is correct?
cwrw238
  • cwrw238
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anonymous
  • anonymous
The roots of a polynomial function are the values of x such that \[f(x) = 0\]
anonymous
  • anonymous
Now, notice that a n-order polynomial can have at most n roots at any time.
anonymous
  • anonymous
:/ I'm confused
cwrw238
  • cwrw238
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
anonymous
  • anonymous
Yes, exactly. I wanted to paint that point in the end.
cwrw238
  • cwrw238
- that way they can both be right
anonymous
  • anonymous
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
anonymous
  • anonymous
First, do you understand that a n-ordered polynomial can have at most n roots, meaning at most n x-axis intercepts?
anonymous
  • anonymous
yes.
anonymous
  • anonymous
Good, then consider the y-intercept. Polynomials must cross the y axis at some point as they are infinitely continuous.
anonymous
  • anonymous
Furthermore, they can only cross the y-axis exactly once.
anonymous
  • anonymous
Otherwise, it will violate the straight line test.
cwrw238
  • cwrw238
- 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 .
anonymous
  • anonymous
therefore only Esmeralda is right because heinz says it only crosses the x three times. not the y at all right?
anonymous
  • anonymous
Not just in the complex case. A 3rd order polynomial can have less x-axis intercepts due to repeated roots.
cwrw238
  • cwrw238
- yep thats true
anonymous
  • anonymous
Okay, so then what is the next step?
anonymous
  • anonymous
To conclude, the answer to your question is that both propositions are correct.
cwrw238
  • cwrw238
No - they are both right because Heinz only refers to the x-axis.
anonymous
  • anonymous
okay ? could y'all help with questions related to this?
anonymous
  • anonymous
Thank you !! @cwrw238 could you help more?
anonymous
  • anonymous
@math1234 could you help more with related question?
cwrw238
  • cwrw238
I would but gotta go right now sorry

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