Ex: Find Key Information about a Given Polynomial Function — Transcript

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1. 0:00In this question, we're given a polynomial  function and asked to provide a variety of
2. 0:05information. We first want to write the  polynomial function in descending order,
3. 0:10which means you want to order the terms  from highest degree to lowest degree.
4. 0:14So looking at all the terms in this polynomial  function, notice how this term here has the
5. 0:20highest degree, so it's going to be the first  term in descending order. The next highest
6. 0:24degree is degree three, so this term would be  second. The next highest degree is degree two,
7. 0:30this would be the third term. This has  degree one, this is the fourth term,
8. 0:35and the constant term has degree zero,  so it's going to be the fifth term.
9. 0:40So in descending order, we would  have f of x equals negative three x
10. 0:47to the sixth plus four x cubed minus  two x squared plus seven x minus three.
11. 1:06Next, we're asked for the degree of the  polynomial function. The degree of the
12. 1:10polynomial function is the degree of the term  with the highest degree. So if we look at the
13. 1:14polynomial function in descending order,  the degree of the polynomial will be the
14. 1:18degree of this first term, which is six.  So the degree of the polynomial is six.
15. 1:25If it's not in descending order, of course,  we'd have to look for the term that has the
16. 1:29highest degree. The leading coefficient is  the coefficient of the term with the highest
17. 1:34degree. Again, if it's in descending order,  we can easily see the leading coefficient is
18. 1:38going to be negative three. The reason  it's called the leading coefficient is
19. 1:43that most of the time we do want to give  polynomial functions in descending order,
20. 1:49so negative three would be the  first or leading coefficient.
21. 1:53Now, the maximum number of real zeros  would be the same as the maximum number
22. 1:57of x-intercepts or horizontal intercepts,  and this is equal to the degree of the
23. 2:01polynomial function. So in this case, the  maximum number of real zeros would be six.
24. 2:08Then, the maximum number of terms  in a polynomial function is equal
25. 2:11to the degree minus one. In this case,  six minus one is equal to five. So the
26. 2:17most number of terms this polynomial  function could have would be five.