Descartes' rule of signs

Lessons

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

• 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:
  $P(x)=2x^6-7x^5+x^4+5x^3-6x^2-10$
  
  

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• 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)

  $P(x)=4x+9x^6-5x^2-10x^7+6x^3-8x^5-7x^4+3$
  
  
  b)

  $P(x)=x^4-5x^2-6x$ (note: NO constant term!!)
  
  

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• 2.
  Use the Rational Roots Theorem, together with Descartes' Rule of Signs, to Find Roots Effectively
  Solve:
  a)

  $-3x^3+22x^2-37x+10=0$
  
  
  b)

  $-3x^3-5x-7x^2+2x^4-3=0$
  
  

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