Originally Posted by
Rich B. Greetings PHckr:
Okay, you can do that. But I did it without this theorem. Further, you made one mistake it has AT MOST two solutions and AT LEAST one, that is the fundamental theorem.
I am perplexed by the first two sentences noted above in blue print. I used one theorem, you used another (MVT). The FTA simplifies the proof with a single stroke of logic. Of course, such statement is meaningless should the remaining print in blue indeed be true. As it were, however, the fundamental theorem of algebra guarantees the existence of exactly n-complex roots for every polynomial of degree, n, having real coefficients. I believe it was Gauss (perhaps Euler --my memory for detail, sadly, varies inversely with age) who first conjectured existence of solutions numbering between one and n (this may be what you had in mind). However, the MVT, established more recently, asserts exactly n. Finally, as I state in my initial dialogue, yes, the plan suggested herein need only be formalized as a proof of, perhaps, three or so lines of text.
BTW, as an alternative to invoking the FTA, Descartes Rule of Signs affirms the singular positive root as well. What's more, such approach obviates any obligation to incorporate solutions into the formal proof.
Regards,
Rich B.