SlideShare uma empresa Scribd logo
1 de 34
Baixar para ler offline
2.5 ZEROS OF POLYNOMIAL FUNCTIONS
Copyright © Cengage Learning. All rights reserved.
2
• Use the Fundamental Theorem of Algebra to
determine the number of zeros of polynomial
functions.
• Find rational zeros of polynomial functions.
• Find conjugate pairs of complex zeros.
• Find zeros of polynomials by factoring.
• Use Descartes’s Rule of Signs and the Upper
and Lower Bound Rules to find zeros of
polynomials.
What You Should Learn
3
The Fundamental Theorem of Algebra
4
The Fundamental Theorem of Algebra
5
Example 1 – Zeros of Polynomial Functions
a. The first-degree polynomial f(x) = x – 2 has exactly one
zero: x = 2.
b. Counting multiplicity, the second-degree polynomial
function
f(x) = x2 – 6x + 9
= (x – 3)(x – 3)
has exactly two zeros: x = 3 and x = 3
(repeated zero)
6
Example 1 – Zeros of Polynomial Functions
c. The third-degree polynomial function
f(x) = x3 + 4x
= x(x2 + 4)
= x(x – 2i)(x + 2i)
has exactly three zeros: x = 0, x = 2i, and x = –2i.
cont’d
7
Example 1 – Zeros of Polynomial Functions
d. The fourth-degree polynomial function
f(x) = x4 – 1
= (x – 1)(x + 1)(x – i)(x + i)
has exactly four zeros: x = 1, x = –1, x = i, and x = –i.
cont’d
8
The Rational Zero Test
9
The Rational Zero Test
10
Example 3 – Rational Zero Test with Leading Coefficient of 1
Find the rational zeros of f(x) = x4 – x3 + x2 – 3x – 6.
Solution:
Because the leading coefficient is 1, the possible rational
zeros are the factors of the constant term.
Possible rational zeros: ±1, ±2, ±3, ±6
11
Example 3 – Solution
So, f(x) factors as
f(x) = (x + 1)(x – 2)(x2 + 3).
0 remainder, so x = –1 is a zero.
0 remainder, so x = 2 is a zero.
cont’d
12
Example 3 – Solution cont’d
Figure 2.32
13
The Rational Zero Test
If the leading coefficient of a polynomial is not 1,
(1) * a programmable calculator
(2) a graph:good estimate of the locations of the zeros;
(3) the Intermediate Value Theorem along with a table
(4) synthetic division can be used to test the possible
rational zeros.
14
Example 5
Find all the real zeros of 12161510)( 23
 xxxxf
15
Conjugate Pairs
16
Conjugate Pairs
The pairs of complex zeros of the form a + bi and a – bi are
conjugates.
17
Example 7 Finding the Zeros of a Polynomial Functions
Find all the zeros of given that
is a zero of f.
60263)( 234
 xxxxxf
i31
18
Example 6 – Finding a Polynomial with Given Zeros
Find a fourth-degree polynomial function with real
coefficients that has –1, –1, and 3i as zeros.
Solution:
Because 3i is a zero and the polynomial is stated to have
real coefficients, conjugate –3i must also be a zero.
So, from the Linear Factorization Theorem, f(x) can be
written as
f(x) = a(x + 1)(x + 1)(x – 3i)(x + 3i).
19
Example 6 – Solution
For simplicity, let a = 1 to obtain
f(x) = (x2 + 2x + 1)(x2 + 9)
= x4 + 2x3 + 10x2 + 18x + 9.
cont’d
20
Factoring a Polynomial
21
Factoring a Polynomial
f(x) = an(x – c1)(x – c2)(x – c3) . . . (x – cn)
22
Factoring a Polynomial
A quadratic factor with no real zeros is said to be prime or
irreducible over the reals.
x2 + 1 = (x – i)(x + i) is irreducible over the reals (and
therefore over the rationals).
x2 – 2 = is irreducible over the rationals
but reducible over the reals.
23
Example 7 – Finding the Zeros of a Polynomial Function
Find all the zeros of f(x) = x4 – 3x3 + 6x2 + 2x – 60 given that
1 + 3i is a zero of f.
Solution:
Because complex zeros occur in conjugate pairs, you know
that 1 – 3i is also a zero of f.
Both [x – (1 + 3i)] and [x – (1 – 3i)] are factors of f.
[x – (1 + 3i)][x – (1 – 3i)]
= [(x – 1) – 3i][(x – 1) + 3i]
= (x – 1)2 – 9i2
= x2 – 2x + 10.
24
Example 7 – Solution cont’d
25
Example 7 – Solution
f(x) = (x2 – 2x + 10)(x2 – x – 6)
= (x2 – 2x + 10)(x – 3)(x + 2)
The zeros of f are x = 1 + 3i, x = 1 – 3i, x = 3, and x = –2.
cont’d
26
Other Tests for Zeros of Polynomials
27
Other Tests for Zeros of Polynomials
A variation in sign means that two consecutive coefficients
have opposite signs.
28
Other Tests for Zeros of Polynomials
When using Descartes’s Rule of Signs, a zero of multiplicity
k should be counted as k zeros.
x3 – 3x + 2 = (x – 1)(x – 1)(x + 2)
the two positive real zeros are x = 1 of multiplicity 2.
29
Example 9 – Using Descartes’s Rule of Signs
Describe the possible real zeros of
f(x) = 3x3 – 5x2 + 6x – 4.
Solution:
The original polynomial has three variations in sign.
f(x) = 3x3 – 5x2 + 6x – 4.
30
Example 9 – Solution
The polynomial
f(–x) = 3(–x)3 – 5(–x)2 + 6(–x) – 4
= –3x3 – 5x2 – 6x – 4
has no variations in sign.
So, from Descartes’s Rule of Signs, the polynomial
f(x) = 3x3 – 5x2 + 6x – 4 has either three positive real zeros
or one positive real zero, and has no negative real zeros.
cont’d
31
Example 9 – Solution cont’d
32
Other Tests for Zeros of Polynomials
33
Other Tests for Zeros of Polynomials
1. If the terms of f(x) have a common monomial factor, it
should be factored out before applying the tests in this
section. For instance, by writing
f(x) = x4 – 5x3 + 3x2 + x
= x(x3 – 5x2 + 3x + 1)
you can see that x = 0 is a zero of f and that the remaining
zeros can be obtained by analyzing the cubic factor.
34
Other Tests for Zeros of Polynomials
2. If you are able to find all but two zeros of f(x), you can
always use the Quadratic Formula on the remaining
quadratic factor. For instance, if you succeeded in writing
f(x) = x4 – 5x3 + 3x2 + x
= x(x – 1)(x2 – 4x – 1)
you can apply the Quadratic Formula to x2 – 4x – 1 to
conclude that the two remaining zeros are x = 2 +
x = 2 – .

Mais conteúdo relacionado

Mais procurados

Polynomials Grade 10
Polynomials Grade 10Polynomials Grade 10
Polynomials Grade 10ingroy
 
Factoring polynomials
Factoring polynomialsFactoring polynomials
Factoring polynomialsMark Ryder
 
2.2 Polynomial Function Notes
2.2 Polynomial Function Notes2.2 Polynomial Function Notes
2.2 Polynomial Function Noteslgemgnani
 
Module 3 polynomial functions
Module 3   polynomial functionsModule 3   polynomial functions
Module 3 polynomial functionsdionesioable
 
Factoring polynomials
Factoring polynomialsFactoring polynomials
Factoring polynomialsPaco Marcos
 
Rational Root Theorem
Rational Root TheoremRational Root Theorem
Rational Root Theoremcmorgancavo
 
Factor Theorem and Remainder Theorem
Factor Theorem and Remainder TheoremFactor Theorem and Remainder Theorem
Factor Theorem and Remainder TheoremRonalie Mejos
 
Long division, synthetic division, remainder theorem and factor theorem
Long division, synthetic division, remainder theorem and factor theoremLong division, synthetic division, remainder theorem and factor theorem
Long division, synthetic division, remainder theorem and factor theoremJohn Rome Aranas
 
Sim(mathematics 10 polynomial functions)
Sim(mathematics 10 polynomial functions) Sim(mathematics 10 polynomial functions)
Sim(mathematics 10 polynomial functions) ileen menes
 
Polynomial Function by Desirae &
Polynomial Function by Desirae & Polynomial Function by Desirae &
Polynomial Function by Desirae & Hope Scott
 
3.3 Zeros of Polynomial Functions
3.3 Zeros of Polynomial Functions3.3 Zeros of Polynomial Functions
3.3 Zeros of Polynomial Functionssmiller5
 
Mat221 5.6 definite integral substitutions and the area between two curves
Mat221 5.6 definite integral substitutions and the area between two curvesMat221 5.6 definite integral substitutions and the area between two curves
Mat221 5.6 definite integral substitutions and the area between two curvesGlenSchlee
 
Notes solving polynomial equations
Notes   solving polynomial equationsNotes   solving polynomial equations
Notes solving polynomial equationsLori Rapp
 
Remainder and factor theorem
Remainder and factor theoremRemainder and factor theorem
Remainder and factor theoremSarah Sue Calbio
 

Mais procurados (20)

Polynomials Grade 10
Polynomials Grade 10Polynomials Grade 10
Polynomials Grade 10
 
Factoring polynomials
Factoring polynomialsFactoring polynomials
Factoring polynomials
 
Remainder theorem
Remainder theoremRemainder theorem
Remainder theorem
 
2.2 Polynomial Function Notes
2.2 Polynomial Function Notes2.2 Polynomial Function Notes
2.2 Polynomial Function Notes
 
Module 3 polynomial functions
Module 3   polynomial functionsModule 3   polynomial functions
Module 3 polynomial functions
 
Polynomial equations
Polynomial equationsPolynomial equations
Polynomial equations
 
Factoring polynomials
Factoring polynomialsFactoring polynomials
Factoring polynomials
 
Rational Root Theorem
Rational Root TheoremRational Root Theorem
Rational Root Theorem
 
Remainder theorem
Remainder theoremRemainder theorem
Remainder theorem
 
Factor Theorem and Remainder Theorem
Factor Theorem and Remainder TheoremFactor Theorem and Remainder Theorem
Factor Theorem and Remainder Theorem
 
Mathematics 10 (Quarter Two)
Mathematics 10 (Quarter Two)Mathematics 10 (Quarter Two)
Mathematics 10 (Quarter Two)
 
Polynomial equations
Polynomial equationsPolynomial equations
Polynomial equations
 
Long division, synthetic division, remainder theorem and factor theorem
Long division, synthetic division, remainder theorem and factor theoremLong division, synthetic division, remainder theorem and factor theorem
Long division, synthetic division, remainder theorem and factor theorem
 
Sim(mathematics 10 polynomial functions)
Sim(mathematics 10 polynomial functions) Sim(mathematics 10 polynomial functions)
Sim(mathematics 10 polynomial functions)
 
Polynomial Function by Desirae &
Polynomial Function by Desirae & Polynomial Function by Desirae &
Polynomial Function by Desirae &
 
3.3 Zeros of Polynomial Functions
3.3 Zeros of Polynomial Functions3.3 Zeros of Polynomial Functions
3.3 Zeros of Polynomial Functions
 
Mat221 5.6 definite integral substitutions and the area between two curves
Mat221 5.6 definite integral substitutions and the area between two curvesMat221 5.6 definite integral substitutions and the area between two curves
Mat221 5.6 definite integral substitutions and the area between two curves
 
0303 ch 3 day 3
0303 ch 3 day 30303 ch 3 day 3
0303 ch 3 day 3
 
Notes solving polynomial equations
Notes   solving polynomial equationsNotes   solving polynomial equations
Notes solving polynomial equations
 
Remainder and factor theorem
Remainder and factor theoremRemainder and factor theorem
Remainder and factor theorem
 

Semelhante a 2 5 zeros of poly fn

Synthetic and Remainder Theorem of Polynomials.ppt
Synthetic and Remainder Theorem of Polynomials.pptSynthetic and Remainder Theorem of Polynomials.ppt
Synthetic and Remainder Theorem of Polynomials.pptMarkVincentDoria1
 
Module 2 exponential functions
Module 2   exponential functionsModule 2   exponential functions
Module 2 exponential functionsdionesioable
 
Factoring Polynomials (1).pptx
Factoring Polynomials (1).pptxFactoring Polynomials (1).pptx
Factoring Polynomials (1).pptxMartiNBaccay2
 
5.5 Zeros of Polynomial Functions
5.5 Zeros of Polynomial Functions5.5 Zeros of Polynomial Functions
5.5 Zeros of Polynomial Functionssmiller5
 
Algebra lesson 4.2 zeroes of quadratic functions
Algebra lesson 4.2 zeroes of quadratic functionsAlgebra lesson 4.2 zeroes of quadratic functions
Algebra lesson 4.2 zeroes of quadratic functionspipamutuc
 
Dividing polynomials
Dividing polynomialsDividing polynomials
Dividing polynomialsEducación
 
Module 3 quadratic functions
Module 3   quadratic functionsModule 3   quadratic functions
Module 3 quadratic functionsdionesioable
 
2 1 polynomials
2 1 polynomials2 1 polynomials
2 1 polynomialshisema01
 
Solving quadratic equations[1]
Solving quadratic equations[1]Solving quadratic equations[1]
Solving quadratic equations[1]RobinFilter
 
Solving quadratic equations
Solving quadratic equationsSolving quadratic equations
Solving quadratic equationssrobbins4
 
Higher Maths 2.1.1 - Polynomials
Higher Maths 2.1.1 - PolynomialsHigher Maths 2.1.1 - Polynomials
Higher Maths 2.1.1 - Polynomialstimschmitz
 
Factoring Polynomials
Factoring PolynomialsFactoring Polynomials
Factoring Polynomialsitutor
 
GR 8 Math Powerpoint about Polynomial Techniques
GR 8 Math Powerpoint about Polynomial TechniquesGR 8 Math Powerpoint about Polynomial Techniques
GR 8 Math Powerpoint about Polynomial Techniquesreginaatin
 
best for me1017103 634411962199405000 (2)
best for me1017103 634411962199405000 (2)best for me1017103 634411962199405000 (2)
best for me1017103 634411962199405000 (2)Sourav Rider
 
Cbse 12 Class Maths Sample Paper
Cbse 12 Class Maths Sample PaperCbse 12 Class Maths Sample Paper
Cbse 12 Class Maths Sample PaperSunaina Rawat
 
5.7 Interactive Classroom Roots and Zeros.ppt
5.7 Interactive Classroom Roots and Zeros.ppt5.7 Interactive Classroom Roots and Zeros.ppt
5.7 Interactive Classroom Roots and Zeros.pptAARow1
 

Semelhante a 2 5 zeros of poly fn (20)

Synthetic and Remainder Theorem of Polynomials.ppt
Synthetic and Remainder Theorem of Polynomials.pptSynthetic and Remainder Theorem of Polynomials.ppt
Synthetic and Remainder Theorem of Polynomials.ppt
 
Module 2 exponential functions
Module 2   exponential functionsModule 2   exponential functions
Module 2 exponential functions
 
Unit2.polynomials.algebraicfractions
Unit2.polynomials.algebraicfractionsUnit2.polynomials.algebraicfractions
Unit2.polynomials.algebraicfractions
 
Factoring Polynomials (1).pptx
Factoring Polynomials (1).pptxFactoring Polynomials (1).pptx
Factoring Polynomials (1).pptx
 
5.5 Zeros of Polynomial Functions
5.5 Zeros of Polynomial Functions5.5 Zeros of Polynomial Functions
5.5 Zeros of Polynomial Functions
 
Algebra lesson 4.2 zeroes of quadratic functions
Algebra lesson 4.2 zeroes of quadratic functionsAlgebra lesson 4.2 zeroes of quadratic functions
Algebra lesson 4.2 zeroes of quadratic functions
 
Dividing polynomials
Dividing polynomialsDividing polynomials
Dividing polynomials
 
Module 3 quadratic functions
Module 3   quadratic functionsModule 3   quadratic functions
Module 3 quadratic functions
 
2 1 polynomials
2 1 polynomials2 1 polynomials
2 1 polynomials
 
Solving quadratic equations[1]
Solving quadratic equations[1]Solving quadratic equations[1]
Solving quadratic equations[1]
 
Solving quadratic equations
Solving quadratic equationsSolving quadratic equations
Solving quadratic equations
 
Higher Maths 2.1.1 - Polynomials
Higher Maths 2.1.1 - PolynomialsHigher Maths 2.1.1 - Polynomials
Higher Maths 2.1.1 - Polynomials
 
Factoring Polynomials
Factoring PolynomialsFactoring Polynomials
Factoring Polynomials
 
GR 8 Math Powerpoint about Polynomial Techniques
GR 8 Math Powerpoint about Polynomial TechniquesGR 8 Math Powerpoint about Polynomial Techniques
GR 8 Math Powerpoint about Polynomial Techniques
 
11.2
11.211.2
11.2
 
Chapter 3
Chapter 3Chapter 3
Chapter 3
 
AP Calculus Project
AP Calculus ProjectAP Calculus Project
AP Calculus Project
 
best for me1017103 634411962199405000 (2)
best for me1017103 634411962199405000 (2)best for me1017103 634411962199405000 (2)
best for me1017103 634411962199405000 (2)
 
Cbse 12 Class Maths Sample Paper
Cbse 12 Class Maths Sample PaperCbse 12 Class Maths Sample Paper
Cbse 12 Class Maths Sample Paper
 
5.7 Interactive Classroom Roots and Zeros.ppt
5.7 Interactive Classroom Roots and Zeros.ppt5.7 Interactive Classroom Roots and Zeros.ppt
5.7 Interactive Classroom Roots and Zeros.ppt
 

Último

UiPath Studio Web workshop series - Day 7
UiPath Studio Web workshop series - Day 7UiPath Studio Web workshop series - Day 7
UiPath Studio Web workshop series - Day 7DianaGray10
 
Nanopower In Semiconductor Industry.pdf
Nanopower  In Semiconductor Industry.pdfNanopower  In Semiconductor Industry.pdf
Nanopower In Semiconductor Industry.pdfPedro Manuel
 
UiPath Community: AI for UiPath Automation Developers
UiPath Community: AI for UiPath Automation DevelopersUiPath Community: AI for UiPath Automation Developers
UiPath Community: AI for UiPath Automation DevelopersUiPathCommunity
 
Linked Data in Production: Moving Beyond Ontologies
Linked Data in Production: Moving Beyond OntologiesLinked Data in Production: Moving Beyond Ontologies
Linked Data in Production: Moving Beyond OntologiesDavid Newbury
 
Meet the new FSP 3000 M-Flex800™
Meet the new FSP 3000 M-Flex800™Meet the new FSP 3000 M-Flex800™
Meet the new FSP 3000 M-Flex800™Adtran
 
IaC & GitOps in a Nutshell - a FridayInANuthshell Episode.pdf
IaC & GitOps in a Nutshell - a FridayInANuthshell Episode.pdfIaC & GitOps in a Nutshell - a FridayInANuthshell Episode.pdf
IaC & GitOps in a Nutshell - a FridayInANuthshell Episode.pdfDaniel Santiago Silva Capera
 
Artificial Intelligence & SEO Trends for 2024
Artificial Intelligence & SEO Trends for 2024Artificial Intelligence & SEO Trends for 2024
Artificial Intelligence & SEO Trends for 2024D Cloud Solutions
 
Machine Learning Model Validation (Aijun Zhang 2024).pdf
Machine Learning Model Validation (Aijun Zhang 2024).pdfMachine Learning Model Validation (Aijun Zhang 2024).pdf
Machine Learning Model Validation (Aijun Zhang 2024).pdfAijun Zhang
 
NIST Cybersecurity Framework (CSF) 2.0 Workshop
NIST Cybersecurity Framework (CSF) 2.0 WorkshopNIST Cybersecurity Framework (CSF) 2.0 Workshop
NIST Cybersecurity Framework (CSF) 2.0 WorkshopBachir Benyammi
 
Crea il tuo assistente AI con lo Stregatto (open source python framework)
Crea il tuo assistente AI con lo Stregatto (open source python framework)Crea il tuo assistente AI con lo Stregatto (open source python framework)
Crea il tuo assistente AI con lo Stregatto (open source python framework)Commit University
 
VoIP Service and Marketing using Odoo and Asterisk PBX
VoIP Service and Marketing using Odoo and Asterisk PBXVoIP Service and Marketing using Odoo and Asterisk PBX
VoIP Service and Marketing using Odoo and Asterisk PBXTarek Kalaji
 
IESVE Software for Florida Code Compliance Using ASHRAE 90.1-2019
IESVE Software for Florida Code Compliance Using ASHRAE 90.1-2019IESVE Software for Florida Code Compliance Using ASHRAE 90.1-2019
IESVE Software for Florida Code Compliance Using ASHRAE 90.1-2019IES VE
 
Building AI-Driven Apps Using Semantic Kernel.pptx
Building AI-Driven Apps Using Semantic Kernel.pptxBuilding AI-Driven Apps Using Semantic Kernel.pptx
Building AI-Driven Apps Using Semantic Kernel.pptxUdaiappa Ramachandran
 
Salesforce Miami User Group Event - 1st Quarter 2024
Salesforce Miami User Group Event - 1st Quarter 2024Salesforce Miami User Group Event - 1st Quarter 2024
Salesforce Miami User Group Event - 1st Quarter 2024SkyPlanner
 
Empowering Africa's Next Generation: The AI Leadership Blueprint
Empowering Africa's Next Generation: The AI Leadership BlueprintEmpowering Africa's Next Generation: The AI Leadership Blueprint
Empowering Africa's Next Generation: The AI Leadership BlueprintMahmoud Rabie
 
Computer 10: Lesson 10 - Online Crimes and Hazards
Computer 10: Lesson 10 - Online Crimes and HazardsComputer 10: Lesson 10 - Online Crimes and Hazards
Computer 10: Lesson 10 - Online Crimes and HazardsSeth Reyes
 
Bird eye's view on Camunda open source ecosystem
Bird eye's view on Camunda open source ecosystemBird eye's view on Camunda open source ecosystem
Bird eye's view on Camunda open source ecosystemAsko Soukka
 
Connector Corner: Extending LLM automation use cases with UiPath GenAI connec...
Connector Corner: Extending LLM automation use cases with UiPath GenAI connec...Connector Corner: Extending LLM automation use cases with UiPath GenAI connec...
Connector Corner: Extending LLM automation use cases with UiPath GenAI connec...DianaGray10
 

Último (20)

20150722 - AGV
20150722 - AGV20150722 - AGV
20150722 - AGV
 
UiPath Studio Web workshop series - Day 7
UiPath Studio Web workshop series - Day 7UiPath Studio Web workshop series - Day 7
UiPath Studio Web workshop series - Day 7
 
Nanopower In Semiconductor Industry.pdf
Nanopower  In Semiconductor Industry.pdfNanopower  In Semiconductor Industry.pdf
Nanopower In Semiconductor Industry.pdf
 
UiPath Community: AI for UiPath Automation Developers
UiPath Community: AI for UiPath Automation DevelopersUiPath Community: AI for UiPath Automation Developers
UiPath Community: AI for UiPath Automation Developers
 
20230104 - machine vision
20230104 - machine vision20230104 - machine vision
20230104 - machine vision
 
Linked Data in Production: Moving Beyond Ontologies
Linked Data in Production: Moving Beyond OntologiesLinked Data in Production: Moving Beyond Ontologies
Linked Data in Production: Moving Beyond Ontologies
 
Meet the new FSP 3000 M-Flex800™
Meet the new FSP 3000 M-Flex800™Meet the new FSP 3000 M-Flex800™
Meet the new FSP 3000 M-Flex800™
 
IaC & GitOps in a Nutshell - a FridayInANuthshell Episode.pdf
IaC & GitOps in a Nutshell - a FridayInANuthshell Episode.pdfIaC & GitOps in a Nutshell - a FridayInANuthshell Episode.pdf
IaC & GitOps in a Nutshell - a FridayInANuthshell Episode.pdf
 
Artificial Intelligence & SEO Trends for 2024
Artificial Intelligence & SEO Trends for 2024Artificial Intelligence & SEO Trends for 2024
Artificial Intelligence & SEO Trends for 2024
 
Machine Learning Model Validation (Aijun Zhang 2024).pdf
Machine Learning Model Validation (Aijun Zhang 2024).pdfMachine Learning Model Validation (Aijun Zhang 2024).pdf
Machine Learning Model Validation (Aijun Zhang 2024).pdf
 
NIST Cybersecurity Framework (CSF) 2.0 Workshop
NIST Cybersecurity Framework (CSF) 2.0 WorkshopNIST Cybersecurity Framework (CSF) 2.0 Workshop
NIST Cybersecurity Framework (CSF) 2.0 Workshop
 
Crea il tuo assistente AI con lo Stregatto (open source python framework)
Crea il tuo assistente AI con lo Stregatto (open source python framework)Crea il tuo assistente AI con lo Stregatto (open source python framework)
Crea il tuo assistente AI con lo Stregatto (open source python framework)
 
VoIP Service and Marketing using Odoo and Asterisk PBX
VoIP Service and Marketing using Odoo and Asterisk PBXVoIP Service and Marketing using Odoo and Asterisk PBX
VoIP Service and Marketing using Odoo and Asterisk PBX
 
IESVE Software for Florida Code Compliance Using ASHRAE 90.1-2019
IESVE Software for Florida Code Compliance Using ASHRAE 90.1-2019IESVE Software for Florida Code Compliance Using ASHRAE 90.1-2019
IESVE Software for Florida Code Compliance Using ASHRAE 90.1-2019
 
Building AI-Driven Apps Using Semantic Kernel.pptx
Building AI-Driven Apps Using Semantic Kernel.pptxBuilding AI-Driven Apps Using Semantic Kernel.pptx
Building AI-Driven Apps Using Semantic Kernel.pptx
 
Salesforce Miami User Group Event - 1st Quarter 2024
Salesforce Miami User Group Event - 1st Quarter 2024Salesforce Miami User Group Event - 1st Quarter 2024
Salesforce Miami User Group Event - 1st Quarter 2024
 
Empowering Africa's Next Generation: The AI Leadership Blueprint
Empowering Africa's Next Generation: The AI Leadership BlueprintEmpowering Africa's Next Generation: The AI Leadership Blueprint
Empowering Africa's Next Generation: The AI Leadership Blueprint
 
Computer 10: Lesson 10 - Online Crimes and Hazards
Computer 10: Lesson 10 - Online Crimes and HazardsComputer 10: Lesson 10 - Online Crimes and Hazards
Computer 10: Lesson 10 - Online Crimes and Hazards
 
Bird eye's view on Camunda open source ecosystem
Bird eye's view on Camunda open source ecosystemBird eye's view on Camunda open source ecosystem
Bird eye's view on Camunda open source ecosystem
 
Connector Corner: Extending LLM automation use cases with UiPath GenAI connec...
Connector Corner: Extending LLM automation use cases with UiPath GenAI connec...Connector Corner: Extending LLM automation use cases with UiPath GenAI connec...
Connector Corner: Extending LLM automation use cases with UiPath GenAI connec...
 

2 5 zeros of poly fn

  • 1. 2.5 ZEROS OF POLYNOMIAL FUNCTIONS Copyright © Cengage Learning. All rights reserved.
  • 2. 2 • Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions. • Find rational zeros of polynomial functions. • Find conjugate pairs of complex zeros. • Find zeros of polynomials by factoring. • Use Descartes’s Rule of Signs and the Upper and Lower Bound Rules to find zeros of polynomials. What You Should Learn
  • 5. 5 Example 1 – Zeros of Polynomial Functions a. The first-degree polynomial f(x) = x – 2 has exactly one zero: x = 2. b. Counting multiplicity, the second-degree polynomial function f(x) = x2 – 6x + 9 = (x – 3)(x – 3) has exactly two zeros: x = 3 and x = 3 (repeated zero)
  • 6. 6 Example 1 – Zeros of Polynomial Functions c. The third-degree polynomial function f(x) = x3 + 4x = x(x2 + 4) = x(x – 2i)(x + 2i) has exactly three zeros: x = 0, x = 2i, and x = –2i. cont’d
  • 7. 7 Example 1 – Zeros of Polynomial Functions d. The fourth-degree polynomial function f(x) = x4 – 1 = (x – 1)(x + 1)(x – i)(x + i) has exactly four zeros: x = 1, x = –1, x = i, and x = –i. cont’d
  • 10. 10 Example 3 – Rational Zero Test with Leading Coefficient of 1 Find the rational zeros of f(x) = x4 – x3 + x2 – 3x – 6. Solution: Because the leading coefficient is 1, the possible rational zeros are the factors of the constant term. Possible rational zeros: ±1, ±2, ±3, ±6
  • 11. 11 Example 3 – Solution So, f(x) factors as f(x) = (x + 1)(x – 2)(x2 + 3). 0 remainder, so x = –1 is a zero. 0 remainder, so x = 2 is a zero. cont’d
  • 12. 12 Example 3 – Solution cont’d Figure 2.32
  • 13. 13 The Rational Zero Test If the leading coefficient of a polynomial is not 1, (1) * a programmable calculator (2) a graph:good estimate of the locations of the zeros; (3) the Intermediate Value Theorem along with a table (4) synthetic division can be used to test the possible rational zeros.
  • 14. 14 Example 5 Find all the real zeros of 12161510)( 23  xxxxf
  • 16. 16 Conjugate Pairs The pairs of complex zeros of the form a + bi and a – bi are conjugates.
  • 17. 17 Example 7 Finding the Zeros of a Polynomial Functions Find all the zeros of given that is a zero of f. 60263)( 234  xxxxxf i31
  • 18. 18 Example 6 – Finding a Polynomial with Given Zeros Find a fourth-degree polynomial function with real coefficients that has –1, –1, and 3i as zeros. Solution: Because 3i is a zero and the polynomial is stated to have real coefficients, conjugate –3i must also be a zero. So, from the Linear Factorization Theorem, f(x) can be written as f(x) = a(x + 1)(x + 1)(x – 3i)(x + 3i).
  • 19. 19 Example 6 – Solution For simplicity, let a = 1 to obtain f(x) = (x2 + 2x + 1)(x2 + 9) = x4 + 2x3 + 10x2 + 18x + 9. cont’d
  • 21. 21 Factoring a Polynomial f(x) = an(x – c1)(x – c2)(x – c3) . . . (x – cn)
  • 22. 22 Factoring a Polynomial A quadratic factor with no real zeros is said to be prime or irreducible over the reals. x2 + 1 = (x – i)(x + i) is irreducible over the reals (and therefore over the rationals). x2 – 2 = is irreducible over the rationals but reducible over the reals.
  • 23. 23 Example 7 – Finding the Zeros of a Polynomial Function Find all the zeros of f(x) = x4 – 3x3 + 6x2 + 2x – 60 given that 1 + 3i is a zero of f. Solution: Because complex zeros occur in conjugate pairs, you know that 1 – 3i is also a zero of f. Both [x – (1 + 3i)] and [x – (1 – 3i)] are factors of f. [x – (1 + 3i)][x – (1 – 3i)] = [(x – 1) – 3i][(x – 1) + 3i] = (x – 1)2 – 9i2 = x2 – 2x + 10.
  • 24. 24 Example 7 – Solution cont’d
  • 25. 25 Example 7 – Solution f(x) = (x2 – 2x + 10)(x2 – x – 6) = (x2 – 2x + 10)(x – 3)(x + 2) The zeros of f are x = 1 + 3i, x = 1 – 3i, x = 3, and x = –2. cont’d
  • 26. 26 Other Tests for Zeros of Polynomials
  • 27. 27 Other Tests for Zeros of Polynomials A variation in sign means that two consecutive coefficients have opposite signs.
  • 28. 28 Other Tests for Zeros of Polynomials When using Descartes’s Rule of Signs, a zero of multiplicity k should be counted as k zeros. x3 – 3x + 2 = (x – 1)(x – 1)(x + 2) the two positive real zeros are x = 1 of multiplicity 2.
  • 29. 29 Example 9 – Using Descartes’s Rule of Signs Describe the possible real zeros of f(x) = 3x3 – 5x2 + 6x – 4. Solution: The original polynomial has three variations in sign. f(x) = 3x3 – 5x2 + 6x – 4.
  • 30. 30 Example 9 – Solution The polynomial f(–x) = 3(–x)3 – 5(–x)2 + 6(–x) – 4 = –3x3 – 5x2 – 6x – 4 has no variations in sign. So, from Descartes’s Rule of Signs, the polynomial f(x) = 3x3 – 5x2 + 6x – 4 has either three positive real zeros or one positive real zero, and has no negative real zeros. cont’d
  • 31. 31 Example 9 – Solution cont’d
  • 32. 32 Other Tests for Zeros of Polynomials
  • 33. 33 Other Tests for Zeros of Polynomials 1. If the terms of f(x) have a common monomial factor, it should be factored out before applying the tests in this section. For instance, by writing f(x) = x4 – 5x3 + 3x2 + x = x(x3 – 5x2 + 3x + 1) you can see that x = 0 is a zero of f and that the remaining zeros can be obtained by analyzing the cubic factor.
  • 34. 34 Other Tests for Zeros of Polynomials 2. If you are able to find all but two zeros of f(x), you can always use the Quadratic Formula on the remaining quadratic factor. For instance, if you succeeded in writing f(x) = x4 – 5x3 + 3x2 + x = x(x – 1)(x2 – 4x – 1) you can apply the Quadratic Formula to x2 – 4x – 1 to conclude that the two remaining zeros are x = 2 + x = 2 – .