Yes, I see that now, thanks. I should have specified, however, that this question was multiple choice with answers as follows -
A) P has 11 real roots
B) P has at most 3 real roots \(\displaystyle \text{Imaginary roots come in even numbers.} \)
\(\displaystyle \text{So potentially there could be an odd number of roots, }\)
\(\displaystyle \text{such as 3 real roots.}\)
C) P has no real roots
D) P has at most 2 real roots\(\displaystyle \text{Because the imaginary roots (when they occur)}\)
\(\displaystyle \text{ come in groups with an even number of roots, then there could }\)
\(\displaystyle \text{not be a polynomial with 2 real roots.}\)
E) P has 10 real roots
Obviously, A, C, and E are thrown out just by graphing it, but I don't really see how you can just look at the equation without graphing it and apply Rolle's Theorem to see that either B or D are the correct answer. When you look at end behavior, the values will get really huge as X gets larger, and much smaller as x gets smaller, but how do you apply the approximation? How can I tell that P has at most 2 or 3 real roots? This book is really irritating me with its explanation, so thanks for helping me clear it up