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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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
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To answer this question, think about how and when a polynomial function crosses the x-axis.
I don't know anything about polynomial functions and have to complete this course this week!
well one of the intercepts could be through the y-axis

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Other answers:

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

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