Determine whether each statement is true or false. If false, expl...

Question

Determine whether each statement is true or false. If false, explain why. A polynomial function having degree 6 and only real coefficients may have no real zeros.

Accepted Answer

Hey, everyone here we are asked to identify whether the given statement is true or not. If it is false, we are then asked to write the reason. Here we are given a polynomial function of degree four and having real coefficients may have no real zeros. Here we have two answer choice options, answer a true and answer B false. So here to solve this problem, we just need to recall all the different possible variations of zeros that our polynomial function of degree four may have. So again, we have real coefficients for a polynomial function with a degree of four. And so since we have a degree of four, we can first recall that we could have four total zeros and all four total zeros could be real zeros. We then can also have the variation where we just have two real zeros, which means the other two zeros must be complex. So we can have four real zeros or two real zeros and two complex zeros. And lastly, we can also have all four be complex zeros. So here we see that, that from this last possibility where we have all of our zeros B complex zeros. We know that our given statement is in fact true because there are instances where a polynomial function of degree four with real coefficient may have no real zeros altogether. So again, our statement is true and therefore we are left with answer choice. A where our statement is true. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

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Determine whether each statement is true or false. If false, explain why. A polynomial function having degree 6 and only real coefficients may have no real zeros.