10.3 Descartes’ Rule of Signs (a) If c. C2, …, Cm are any m nonzero real…

10.3 Descartes’ Rule of Signs (a) If c. C2, …, Cm are any m nonzero real…

10.3 Descartes’ Rule of Signs (a) If c. C2, …, Cm are any m nonzero real numbers, and if 2 consecutive terms of this sequence have opposite signs, we say that these 2 terms present a variation of sign. With this concept, we may state Descartes’ rule of signs, a proof of which may be found in any textbook on the theory of equations, as follows: Let f(x) = 0 be a polynomial equation with real coefficients and arranged in descending powers of x. The number of positive roots of the equation is either equal to the number of variations of signs presented by the coefficients of f(x), or less than this number of variations by a positive even number. The number of negative roots is either equal to the number of variations of signs presented by the coefficients of f(-x), or less than this number of variations by a positive even number. A root of multiplicity mis counted as m roots. Investigate the nature of the roots of the following equations by means of Descartes’ rule of signs: 1. x + 3×8 – 5×3 + 4x + 6 = 0, 2. 2.×7 – 3×4 – x3 – 5 = 0, 3. 3×4 + 10×2 + 5x – 4 = 0. (b) Show that xn – 1 = 0 has exactly 2 real roots if n is even, and only 1 real root if n is odd. (c) Show that x5 + x2 + 1 = 0 has 4 imaginary roots. (d) Prove that if p and q are real and q = 0, the equation x3 + px + q = 0 has 2 imaginary roots when p is positive. (e) Prove that if the roots of a polynomial equation are all positive, the signs of the coefficients are alternately positive and negative.

Calculate the price of your order

550 words
We'll send you the first draft for approval by September 11, 2018 at 10:52 AM
Total price:
$26
The price is based on these factors:
Academic level
Number of pages
Urgency
Basic features
  • Free title page and bibliography
  • Unlimited revisions
  • Plagiarism-free guarantee
  • Money-back guarantee
  • 24/7 support
On-demand options
  • Writer’s samples
  • Part-by-part delivery
  • Overnight delivery
  • Copies of used sources
  • Expert Proofreading
Paper format
  • 275 words per page
  • 12 pt Arial/Times New Roman
  • Double line spacing
  • Any citation style (APA, MLA, Chicago/Turabian, Harvard)

Our guarantees

Delivering a high-quality product at a reasonable price is not enough anymore.
That’s why we have developed 5 beneficial guarantees that will make your experience with our service enjoyable, easy, and safe.

Money-back guarantee

You have to be 100% sure of the quality of your product to give a money-back guarantee. This describes us perfectly. Make sure that this guarantee is totally transparent.

Read more

Zero-plagiarism guarantee

Each paper is composed from scratch, according to your instructions. It is then checked by our plagiarism-detection software. There is no gap where plagiarism could squeeze in.

Read more

Free-revision policy

Thanks to our free revisions, there is no way for you to be unsatisfied. We will work on your paper until you are completely happy with the result.

Read more

Privacy policy

Your email is safe, as we store it according to international data protection rules. Your bank details are secure, as we use only reliable payment systems.

Read more

Fair-cooperation guarantee

By sending us your money, you buy the service we provide. Check out our terms and conditions if you prefer business talks to be laid out in official language.

Read more