Descartes' rule of signs

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  1. Introduction to Descartes' Rule of Signs
  2. Fundamental Theorem of Algebra VS. Descartes' Rule of Signs
  3. Descartes' Rule of Signs – explained.
    exercise: Use Descartes' Rule of Signs to determine the possible combinations of roots of:



  1. Use Descartes' Rule of Signs to Determine the Number of Positive and Negative Roots
    Use Descartes' Rule of Signs to determine the possible number of positive roots and negative roots:
    1. P(x)=4x+9x65x210x7+6x38x57x4+3 P(x)=4x+9x^6-5x^2-10x^7+6x^3-8x^5-7x^4+3
    2. P(x)=x45x26x P(x)=x^4-5x^2-6x (note: NO constant term!!)
  2. Use the Rational Roots Theorem, together with Descartes' Rule of Signs, to Find Roots Effectively
    1. 3x3+22x237x+10=0 -3x^3+22x^2-37x+10=0
    2. 3x35x7x2+2x43=0 -3x^3-5x-7x^2+2x^4-3=0

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Topic Basics
Descartes' Rule of Signs For a polynomial P(x)P(x):
\bullet the number of positive roots = the number of sign changes in P(x)P(x), or less than the sign changes by a multiple of 2.
\bullet the number of negative roots = the number of sign changes in P(x)P(-x), or less than the sign changes by a multiple of 2.

Note: Before applying the Descartes' Rule of Signs, make sure to arrange the terms of the polynomial in descending order of exponents.

trick of Descates' rule of signs