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Descartes' rule of signs
- Intro Lesson: a9:40
- Intro Lesson: b28:38
- Lesson: 1a16:24
- Lesson: 1b27:13
- Lesson: 2a26:26
- Lesson: 2b28:28
Descartes' rule of signs
Lessons
Descartes' Rule of Signs
For a polynomial P(x):
∙ the number of positive roots = the number of sign changes in P(x), or less than the sign changes by a multiple of 2.
∙ 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.
∙ the number of positive roots = the number of sign changes in P(x), or less than the sign changes by a multiple of 2.
∙ 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.

- IntroductionIntroduction to Descartes' Rule of Signsa)Fundamental Theorem of Algebra VS. Descartes' Rule of Signsb)Descartes' Rule of Signs – explained.
exercise: Use Descartes' Rule of Signs to determine the possible combinations of roots of:
P(x)=2x6−7x5+x4+5x3−6x2−10 - 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+9x6−5x2−10x7+6x3−8x5−7x4+3b)P(x)=x4−5x2−6x (note: NO constant term!!) - 2.Use the Rational Roots Theorem, together with Descartes' Rule of Signs, to Find Roots Effectively
Solve:a)−3x3+22x2−37x+10=0b)−3x3−5x−7x2+2x4−3=0
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3.
Polynomial Functions
3.1
What is a polynomial function?
3.2
Polynomial long division
3.3
Polynomial synthetic division
3.4
Remainder theorem
3.5
Factor theorem
3.6
Rational zero theorem
3.7
Characteristics of polynomial graphs
3.8
Multiplicities of polynomials
3.9
Imaginary zeros of polynomials
3.10
Determining the equation of a polynomial function
3.11
Applications of polynomial functions
3.12
Solving polynomial inequalities
3.13
Fundamental theorem of algebra
3.14
Descartes' rule of signs