Descartes' rule of signs

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Introduction

Lessons

  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:
    P(x)=2x67x5+x4+5x36x210P(x)=2x^6-7x^5+x^4+5x^3-6x^2-10
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Examples

Lessons

  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
    Solve:
    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