Descartes’ rule of signs

Descartes’ rule of signs

Lessons

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
  • Introduction
    Introduction to Descartes’ Rule of Signs
    a)
    Fundamental Theorem of Algebra VS. Descartes’ Rule of Signs

    b)
    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


  • 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:
    a)
    P(x)=4x+9x65x210x7+6x38x57x4+3 P(x)=4x+9x^6-5x^2-10x^7+6x^3-8x^5-7x^4+3

    b)
    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:
    a)
    3x3+22x237x+10=0 -3x^3+22x^2-37x+10=0

    b)
    3x35x7x2+2x43=0 -3x^3-5x-7x^2+2x^4-3=0