The document discusses applications of factoring polynomials. It provides examples of how factoring can be used to evaluate polynomials by substituting values into the factored form. Factoring is also useful for determining the sign of outputs and for solving polynomial equations, which is described as the most important application of factoring. Examples are given to demonstrate evaluating polynomials both with and without factoring, and checking the answers obtained from factoring using the expanded form.
Rational exponents can be written in three different forms. To evaluate or simplify expressions with rational exponents:
- Use properties of exponents like power-to-a-power, product-to-a-power, and quotient-to-a-power laws
- Simplify to remove negative exponents, fractional exponents in the denominator, or complex fractions
- Write expressions with rational exponents as radicals and simplify if possible in 1-2 sentences
Solving problems involving linear inequalities in two variablesAnirach Ytirahc
This document discusses solving problems involving linear inequalities in two variables. It begins by stating the learning objectives, which are to solve such problems and appreciate their use in real-life situations. An activity is presented involving using a budget to purchase ingredients for chicken adobo. Students are asked to model word problems using linear inequalities with two variables. Examples are provided and students practice translating situations into inequalities. The document solves sample problems and discusses using inequalities to represent real-life scenarios. Students are ultimately tasked with finding examples of such situations from their own experiences.
If you are looking for math video tutorials (with voice recording), you may download it on our YouTube Channel. Don't forget to SUBSCRIBE for you to get updated on our upcoming videos.
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This document discusses solving quadratic inequalities. It provides examples of single-variable quadratic inequalities and explains how to find the solution set by first setting the inequality equal to an equation, solving for the roots, and then testing values within the intervals formed by the roots. The document also introduces quadratic inequalities with two variables and how to represent them. It defines a quadratic inequality and explains the process of solving them by relating it back to solving a quadratic equation.
The document provides a formula and examples for finding the square of a trinomial. The formula is: F2 + M2 + L2 + 2FM + 2FL + 2ML, where F is the first term, M is the middle term, L is the last term, and the other terms represent twice the product of the terms. It then works through four examples applying the formula to find the square of trinomials such as (a + b + c)2, (x + y - 3)2, (x - 2y - 1)2, and (2a - b + b2)2.
Solving Word Problems Involving Quadratic Equationskliegey524
This document provides instructions for solving word problems involving quadratic equations. It explains how to write let statements and equations, solve for consecutive integers or areas, and check solutions. Sample problems are worked through, such as finding two consecutive integers whose sum is 13, or the dimensions of a rectangular garden with an area of 27 square units.
This document provides information about radicals and working with radical expressions. It defines square roots, principal and negative square roots, radicands, perfect squares, cube roots, nth roots, and the product, quotient, and power rules for radicals. It discusses simplifying radical expressions using these rules as well as adding, subtracting, multiplying, and dividing radicals. The document also covers rationalizing denominators, solving radical equations, and using the Pythagorean theorem and distance formula.
Rational exponents can be written in three different forms. To evaluate or simplify expressions with rational exponents:
- Use properties of exponents like power-to-a-power, product-to-a-power, and quotient-to-a-power laws
- Simplify to remove negative exponents, fractional exponents in the denominator, or complex fractions
- Write expressions with rational exponents as radicals and simplify if possible in 1-2 sentences
Solving problems involving linear inequalities in two variablesAnirach Ytirahc
This document discusses solving problems involving linear inequalities in two variables. It begins by stating the learning objectives, which are to solve such problems and appreciate their use in real-life situations. An activity is presented involving using a budget to purchase ingredients for chicken adobo. Students are asked to model word problems using linear inequalities with two variables. Examples are provided and students practice translating situations into inequalities. The document solves sample problems and discusses using inequalities to represent real-life scenarios. Students are ultimately tasked with finding examples of such situations from their own experiences.
If you are looking for math video tutorials (with voice recording), you may download it on our YouTube Channel. Don't forget to SUBSCRIBE for you to get updated on our upcoming videos.
https://tinyurl.com/y9muob6q
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https://tinyurl.com/ycjp8r7u
https://tinyurl.com/ybo27k2u
This document discusses solving quadratic inequalities. It provides examples of single-variable quadratic inequalities and explains how to find the solution set by first setting the inequality equal to an equation, solving for the roots, and then testing values within the intervals formed by the roots. The document also introduces quadratic inequalities with two variables and how to represent them. It defines a quadratic inequality and explains the process of solving them by relating it back to solving a quadratic equation.
The document provides a formula and examples for finding the square of a trinomial. The formula is: F2 + M2 + L2 + 2FM + 2FL + 2ML, where F is the first term, M is the middle term, L is the last term, and the other terms represent twice the product of the terms. It then works through four examples applying the formula to find the square of trinomials such as (a + b + c)2, (x + y - 3)2, (x - 2y - 1)2, and (2a - b + b2)2.
Solving Word Problems Involving Quadratic Equationskliegey524
This document provides instructions for solving word problems involving quadratic equations. It explains how to write let statements and equations, solve for consecutive integers or areas, and check solutions. Sample problems are worked through, such as finding two consecutive integers whose sum is 13, or the dimensions of a rectangular garden with an area of 27 square units.
This document provides information about radicals and working with radical expressions. It defines square roots, principal and negative square roots, radicands, perfect squares, cube roots, nth roots, and the product, quotient, and power rules for radicals. It discusses simplifying radical expressions using these rules as well as adding, subtracting, multiplying, and dividing radicals. The document also covers rationalizing denominators, solving radical equations, and using the Pythagorean theorem and distance formula.
You will learn how to get the value of a, b and c given a quadratic equations.
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This document summarizes a module on rational exponents and radicals that was presented at a 2014 mid-year inset for secondary mathematics teachers. The module covered lessons on zero, negative integral and rational exponents, radicals, and solving radical equations. It provided examples of simplifying expressions using laws of exponents and radicals. Recommended teaching strategies included problem-solving activities and a group brainstorming activity to discuss critical content areas and difficulties from teacher and student perspectives.
You will learn how to evaluate algebraic expressions by substitution.
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Mathematics 9 Lesson 1-A: Solving Quadratic Equations by Completing the SquareJuan Miguel Palero
This powerpoint presentation discusses or talks about the topic or lesson Solving Quadratic Equations using Completing the Square. It also discusses the steps in solving quadratic equations using the method of Completing the Square.
This document provides instruction on evaluating rational algebraic expressions. It begins with learning objectives and a review of translating verbal phrases to rational expressions. It then presents an example of evaluating expressions when variables are given values. The document gives an example problem about a teacher printing modules for students and evaluating the rational expression to determine the number of pages and students. It defines rational algebraic expressions and the steps for evaluating them. Finally, it provides examples of evaluating expressions and an activity for the learner to complete. It emphasizes not dividing by zero which would result in an undefined expression.
The document provides instructions for students on exponential equations and inequalities. It defines exponential equations as equations with variable exponents and exponential inequalities as inequalities with exponents that can be solved similarly to equations. Examples of each are provided along with steps to solve them. Students are given problems to solve and informed that an assessment must be completed by midnight.
This document discusses finding the sum and product of the roots of quadratic equations. It provides the formulas for calculating the sum and product of roots without explicitly solving for the individual roots. The sum of the roots is equal to -b/a, and the product of the roots is equal to c/a, where a, b, and c are the coefficients of the quadratic equation in the standard form ax^2 + bx + c = 0. Several examples are worked out applying these formulas. The document also contains exercises asking the reader to find the sum and product of roots for additional quadratic equations using the given coefficients a, b, and c.
This document discusses the structure of an axiomatic system in mathematics. An axiomatic system consists of undefined terms, defined terms, axioms or postulates, and theorems. Axioms or postulates are statements that are assumed to be true without proof, while theorems are statements that need to be proven using the axioms and previously proven theorems. The document also defines an axiomatic system as a set of axioms used to derive theorems in order to develop and prove statements in a field of mathematics like geometry.
Rational exponents allow radicals to be written as fractional exponents. To write a radical with an index of n as a rational exponent, the index becomes the denominator and the exponent of the radicand becomes the numerator. The rules of exponents still apply to rational exponents. Expressions with rational exponents are simplified by having no negative exponents, no fractional exponents in the denominator, not being a complex fraction, and having the least possible index for any remaining radicals. Examples show how to evaluate, simplify, and perform operations on expressions with rational exponents.
solving quadratic equations using quadratic formulamaricel mas
The document discusses how to use the quadratic formula to solve quadratic equations. It provides the formula: x = (-b ± √(b2 - 4ac)) / 2a. It then works through examples of writing quadratic equations in standard form (ax2 + bx + c = 0) and using the formula to solve them. Specifically, it solves the equations: 1) 1x2 + 3x - 27 = 0, 2) 2x2 + 7x + 5 = 0, 3) x2 - 2x = 8, and 4) x2 - 7x = 10. It concludes by providing 3 additional equations to solve using the quadratic formula.
The document discusses the formulas for summing and subtracting cubes. It presents the SOAP rule - Sum Of APs, for expanding and factorizing expressions involving sums and differences of cubes. Some examples are given to demonstrate applying the rule, such as expanding y + 8a and factorizing s - st.
The document discusses finding the square of a binomial expression by using the FOIL method. It explains that squaring a binomial results in a trinomial with the square of the first term, twice the product of the terms, and the square of the last term. Examples are provided of squaring binomial expressions with variables to demonstrate this perfect square trinomial pattern.
This document contains a 20-question math quiz with questions ranging from easy to difficult on topics of algebra, linear equations, systems of linear equations, and factoring expressions. The quiz is divided into three sections - Easy Round with 5 questions worth 2 points each, Average Round with 6 questions worth 6 points each, and Difficult Round with 10 questions worth 10 points each.
If you are looking for math video tutorials (with voice recording), you may download it on our YouTube Channel. Don't forget to SUBSCRIBE for you to get updated on our upcoming videos.
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This document provides an overview of quadratic equations, beginning with examples of linear and quadratic equations. It defines the standard form of a quadratic equation as ax2 + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. The document also explains the parts of a quadratic equation in standard form and provides additional examples of rewriting equations in standard form.
This document discusses quadratic equations. It defines a quadratic equation as an equation of degree 2 that can be written in the form ax2 + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. It provides examples of complete and incomplete quadratic equations. It also shows how to identify if an equation is quadratic or not and how to transform equations into standard form (ax2 + bx + c = 0) in order to identify the values of a, b, and c.
3 2 solving systems of equations (elimination method)Hazel Joy Chong
The document describes the elimination method for solving systems of equations. The key steps are:
1) Write both equations in standard form Ax + By = C
2) Determine which variable to eliminate using addition or subtraction
3) Solve the resulting equation for one variable
4) Substitute back into the original equation to solve for the other variable
5) Check that the solution satisfies both original equations
It provides examples showing how to set up and solve systems of equations using elimination, including word problems about supplementary angles and finding two numbers based on their sum and difference.
Solving quadratics by completing the squareswartzje
The document discusses solving quadratic equations by completing the square. It provides examples of perfect square trinomials and how to find the missing constant term to create them. The steps for solving a quadratic equation by completing the square are described: 1) move all terms to the left side, 2) find and add the "completing the square" term, 3) factor into a perfect square trinomial, 4) take the square root of each side, 5) solve the resulting equations. Additional examples demonstrate applying these steps to solve quadratic equations algebraically.
The document discusses Cisco Nexus 1000V and the Nexus 1010 appliance. It provides an overview of the Nexus 1000V architecture, comparing it to a physical modular switch. It describes how the Nexus 1000V uses Virtual Supervisor Modules (VSMs) and Virtual Ethernet Modules (VEMs) to replace the functionality of physical linecards and supervisors. It also discusses how the Nexus 1010 appliance allows hosting of VSMs on a physical device for improved performance and redundancy.
You will learn how to get the value of a, b and c given a quadratic equations.
For more instructional resources, CLICK me here! 👇👇👇
https://tinyurl.com/y9muob6q
LIKE and FOLLOW me here! 👍👍👍
https://tinyurl.com/ycjp8r7u
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This document summarizes a module on rational exponents and radicals that was presented at a 2014 mid-year inset for secondary mathematics teachers. The module covered lessons on zero, negative integral and rational exponents, radicals, and solving radical equations. It provided examples of simplifying expressions using laws of exponents and radicals. Recommended teaching strategies included problem-solving activities and a group brainstorming activity to discuss critical content areas and difficulties from teacher and student perspectives.
You will learn how to evaluate algebraic expressions by substitution.
For more instructional resources, CLICK me here! 👇👇👇
https://tinyurl.com/y9muob6q
LIKE and FOLLOW me here! 👍👍👍
https://tinyurl.com/ycjp8r7u
https://tinyurl.com/ybo27k2u
Mathematics 9 Lesson 1-A: Solving Quadratic Equations by Completing the SquareJuan Miguel Palero
This powerpoint presentation discusses or talks about the topic or lesson Solving Quadratic Equations using Completing the Square. It also discusses the steps in solving quadratic equations using the method of Completing the Square.
This document provides instruction on evaluating rational algebraic expressions. It begins with learning objectives and a review of translating verbal phrases to rational expressions. It then presents an example of evaluating expressions when variables are given values. The document gives an example problem about a teacher printing modules for students and evaluating the rational expression to determine the number of pages and students. It defines rational algebraic expressions and the steps for evaluating them. Finally, it provides examples of evaluating expressions and an activity for the learner to complete. It emphasizes not dividing by zero which would result in an undefined expression.
The document provides instructions for students on exponential equations and inequalities. It defines exponential equations as equations with variable exponents and exponential inequalities as inequalities with exponents that can be solved similarly to equations. Examples of each are provided along with steps to solve them. Students are given problems to solve and informed that an assessment must be completed by midnight.
This document discusses finding the sum and product of the roots of quadratic equations. It provides the formulas for calculating the sum and product of roots without explicitly solving for the individual roots. The sum of the roots is equal to -b/a, and the product of the roots is equal to c/a, where a, b, and c are the coefficients of the quadratic equation in the standard form ax^2 + bx + c = 0. Several examples are worked out applying these formulas. The document also contains exercises asking the reader to find the sum and product of roots for additional quadratic equations using the given coefficients a, b, and c.
This document discusses the structure of an axiomatic system in mathematics. An axiomatic system consists of undefined terms, defined terms, axioms or postulates, and theorems. Axioms or postulates are statements that are assumed to be true without proof, while theorems are statements that need to be proven using the axioms and previously proven theorems. The document also defines an axiomatic system as a set of axioms used to derive theorems in order to develop and prove statements in a field of mathematics like geometry.
Rational exponents allow radicals to be written as fractional exponents. To write a radical with an index of n as a rational exponent, the index becomes the denominator and the exponent of the radicand becomes the numerator. The rules of exponents still apply to rational exponents. Expressions with rational exponents are simplified by having no negative exponents, no fractional exponents in the denominator, not being a complex fraction, and having the least possible index for any remaining radicals. Examples show how to evaluate, simplify, and perform operations on expressions with rational exponents.
solving quadratic equations using quadratic formulamaricel mas
The document discusses how to use the quadratic formula to solve quadratic equations. It provides the formula: x = (-b ± √(b2 - 4ac)) / 2a. It then works through examples of writing quadratic equations in standard form (ax2 + bx + c = 0) and using the formula to solve them. Specifically, it solves the equations: 1) 1x2 + 3x - 27 = 0, 2) 2x2 + 7x + 5 = 0, 3) x2 - 2x = 8, and 4) x2 - 7x = 10. It concludes by providing 3 additional equations to solve using the quadratic formula.
The document discusses the formulas for summing and subtracting cubes. It presents the SOAP rule - Sum Of APs, for expanding and factorizing expressions involving sums and differences of cubes. Some examples are given to demonstrate applying the rule, such as expanding y + 8a and factorizing s - st.
The document discusses finding the square of a binomial expression by using the FOIL method. It explains that squaring a binomial results in a trinomial with the square of the first term, twice the product of the terms, and the square of the last term. Examples are provided of squaring binomial expressions with variables to demonstrate this perfect square trinomial pattern.
This document contains a 20-question math quiz with questions ranging from easy to difficult on topics of algebra, linear equations, systems of linear equations, and factoring expressions. The quiz is divided into three sections - Easy Round with 5 questions worth 2 points each, Average Round with 6 questions worth 6 points each, and Difficult Round with 10 questions worth 10 points each.
If you are looking for math video tutorials (with voice recording), you may download it on our YouTube Channel. Don't forget to SUBSCRIBE for you to get updated on our upcoming videos.
https://tinyurl.com/y9muob6q
Also, please do visit our page, LIKE and FOLLOW us on Facebook!
https://tinyurl.com/ycjp8r7u
https://tinyurl.com/ybo27k2u
This document provides an overview of quadratic equations, beginning with examples of linear and quadratic equations. It defines the standard form of a quadratic equation as ax2 + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. The document also explains the parts of a quadratic equation in standard form and provides additional examples of rewriting equations in standard form.
This document discusses quadratic equations. It defines a quadratic equation as an equation of degree 2 that can be written in the form ax2 + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. It provides examples of complete and incomplete quadratic equations. It also shows how to identify if an equation is quadratic or not and how to transform equations into standard form (ax2 + bx + c = 0) in order to identify the values of a, b, and c.
3 2 solving systems of equations (elimination method)Hazel Joy Chong
The document describes the elimination method for solving systems of equations. The key steps are:
1) Write both equations in standard form Ax + By = C
2) Determine which variable to eliminate using addition or subtraction
3) Solve the resulting equation for one variable
4) Substitute back into the original equation to solve for the other variable
5) Check that the solution satisfies both original equations
It provides examples showing how to set up and solve systems of equations using elimination, including word problems about supplementary angles and finding two numbers based on their sum and difference.
Solving quadratics by completing the squareswartzje
The document discusses solving quadratic equations by completing the square. It provides examples of perfect square trinomials and how to find the missing constant term to create them. The steps for solving a quadratic equation by completing the square are described: 1) move all terms to the left side, 2) find and add the "completing the square" term, 3) factor into a perfect square trinomial, 4) take the square root of each side, 5) solve the resulting equations. Additional examples demonstrate applying these steps to solve quadratic equations algebraically.
The document discusses Cisco Nexus 1000V and the Nexus 1010 appliance. It provides an overview of the Nexus 1000V architecture, comparing it to a physical modular switch. It describes how the Nexus 1000V uses Virtual Supervisor Modules (VSMs) and Virtual Ethernet Modules (VEMs) to replace the functionality of physical linecards and supervisors. It also discusses how the Nexus 1010 appliance allows hosting of VSMs on a physical device for improved performance and redundancy.
This document summarizes the key aspects of deduplication on NetApp storage arrays. It discusses what deduplication does, the core enabling technology of fingerprints, how fingerprints work, dedupe metadata space requirements, how dedupe metadata is handled in different ONTAP versions, potential dedupe savings rates for different data types, and considerations around when dedupe is appropriate.
The document introduces the payroll parallel testing process for an ADP implementation. It explains that a parallel test involves loading payroll data from a past period into the new ADP system and comparing the results to the legacy system to validate configurations and ensure accurate payroll processing. It provides an overview of the parallel testing process, including preparation activities, testing scope, and exit criteria to sign off on the implementation. Employees are asked to actively participate and provide input to help ensure a successful transition to the new ADP payroll system.
The document discusses population pharmacokinetic (PK) analysis. Population PK seeks to identify factors that cause variability in drug concentrations among patients and quantify their effects to help determine appropriate dosages. It describes common PK parameters, software used for PK analysis like NONMEM, and approaches for analyzing population PK data, including nonlinear mixed-effects modeling. An example population PK analysis is provided using simulated gentamicin concentration-time data from 30 patients to illustrate modeling the typical response, heterogeneity between individuals, and uncertainty in the model.
The mobile space is exploding and affiliates need to avoid mistakes and move quickly to generate revenue. Discuss practical case studies and learn from our real-life client examples.
Experience level: Beginner, Intermediate, Advanced
Target audience: Affiliate/Publisher, Merchant/Advertiser
Niche/vertical: Mobile
Jeff Stevens, Director of Sales, DirectTrack (Twitter @jstevnz )
If you’re responsible for creating diverse, scalable automated tests but don’t have the time, budget, or a skilled-enough team to create yet another custom test automation framework, then you need to know about Robot Framework!
In this webinar, Bryan Lamb (Founder, RobotFrameworkTutorial.com) and Chris Broesamle (Solutions Engineer, Sauce Labs) will reveal how you can use this powerful, free, open source, generic framework to create continuous automated regression tests for web, batch, API, or database testing. With the simplicity of Robot Framework, in conjunction with Sauce Labs, you can improve your test coverage and time to delivery of your applications.
The document discusses the use of procurement analytics. It begins by explaining what procurement analytics is and why organizations should use it. Analytics can increase demand forecasting accuracy and contract negotiation power. The document then discusses how analytics can be applied in areas like vendor evaluation, spend analysis, and demand forecasting. It also outlines challenges to implementation and provides recommendations for next steps like gaining leadership support, collaborating cross-functionally, developing skills, and integrating systems.
Stylus is a CSS preprocessor that aims to simplify CSS by removing syntactic sugar like brackets and semicolons, enforcing indentation, and allowing variables, mixins, and nested selectors. Nib is a library of mixins, utilities, and components for Stylus that handles vendor prefixes and provides things like clearfixes and hiding text. Together, Stylus and Nib allow for more concise and maintainable CSS code.
This document provides an overview of IBM's Identity and Access Management (IAM) product portfolio, including IBM Security Identity Manager, IBM Security Privileged Identity Manager, and IBM Security Access Manager. It discusses how these products help customers secure access, streamline user provisioning and access requests, safeguard access in cloud/SaaS environments, address compliance needs, and centrally manage privileged identities. Specific capabilities highlighted include identity lifecycle management, self-service access requests, centralized password management, account reconciliation, access recertification, reporting for audits, and broad application integration.
Bridgestone proposes launching run-flat motorcycle tires called Battlax RF in Thailand. Battlax RF provides safety, comfort and fuel efficiency. They allow riders to continue traveling after a tire pressure loss. Thailand has many unpaved roads and motorcycle accidents, creating a potential market. Financial projections show the 5-year NPV is $57.8 million and 10-year is $125.5 million. Battlax RF faces minimal competition and leverages Bridgestone's existing production and distribution in Thailand. The proposal concludes Battlax RF is a lucrative opportunity in line with Bridgestone's mission.
The document discusses the importance of data storage for large tech companies and the challenges of storing large amounts of data reliably. It provides an overview of NetApp's storage solutions, including Data ONTAP, WAFL file system, Snapshot technology, replication tools like SnapMirror, and management tools like My AutoSupport. NetApp believes in providing a unified storage platform with integrated data protection, management and optimization capabilities.
Management 315: International Management, Professor In Hyeock Lee
Loyola University Chicago Spring 2013
This case study analyzes Honda's overall performance as a multinational enterprise using the company's revenue data, 4 distances, firm specific advantages, country specific advantages, foreign direct investment, and much more.
Sham Hassan Chikkegowda, CS Engineer, and Timothee Maret, Senior Developer, of Adobe provide a review of using Security Assertion Markup Language (SAML) with your Experience Manager deployments. SAML is an XML-based, open-standard data format for exchanging authentication and authorization data between parties, in particular, between an identity provider and a service provider. SAML is a product of the OASIS Security Services Technical Committee. To watch the session on demand at http://bit.ly/AEMGems72016 or the MP4 version http://bit.ly/AEMGem72016
HORMONAL REGULATION OF OVULATION,PREGNANCY,PARTURITIONSudarshan Gokhale
The document discusses the hormonal regulation of ovulation, pregnancy, and parturition. It describes the key hormones involved in each process, including estrogen, progesterone, LH, FSH, hCG, relaxin, corticotropin, and oxytocin. Ovulation is regulated by the hypothalamus and pituitary gland releasing hormones like LH and FSH. Pregnancy involves changes in the maternal body and is maintained by hormones like estrogen, progesterone, hCG, and corticotropin. Parturition is triggered by a drop in progesterone and rise in oxytocin, relaxing ligaments and stimulating uterine contractions.
This document summarizes the history and types of surgical dressings. It discusses how dressings have evolved from simple cloths to advanced engineered skin substitutes. The key types of dressings covered are dry dressings, moisture-keeping dressings, bioactive dressings, and skin substitutes. Examples of commonly used dressings like gauze, foams, hydrocolloids, and alginates are provided along with their characteristics and uses.
This document discusses precancerous lesions and conditions of the oral cavity. It defines precancerous lesions as morphologically altered tissues with a higher risk of developing cancer, and precancerous conditions as generalized states associated with significantly increased cancer risk. Examples of precancerous lesions include leukoplakia, erythroplakia, and speckled leukoplakia. Oral submucous fibrosis and sideropenic dysphagia are examples of precancerous conditions. Risk factors for oral cancer development from these lesions include tobacco, alcohol, infections and nutritional deficiencies. Screening, early identification and treatment can help prevent progression to oral squamous cell carcinoma.
3 solving 2nd degree equations by factoring xcTzenma
The document discusses applications of factoring expressions. The main purposes of factoring an expression E into a product E=AB is to utilize properties of multiplication. The most important application of factoring is solving polynomial equations by using the zero-product property. Factoring polynomials makes it easier to evaluate them for given inputs and determine the sign of outputs. Examples are provided to illustrate solving equations by factoring and evaluating polynomials in factored form.
Factoring 15.3 and 15.4 Grouping and Trial and Errorswartzje
This document discusses various methods for factoring trinomials of the form Ax^2 + Bx + C. It begins by outlining three main methods: trial and error, factoring by grouping, and the box method. It then provides examples of using the factoring by grouping method, demonstrating how to find two numbers whose product is AC and sum is B. The document also covers special cases like factoring the difference of squares using the form A^2 - B^2 = (A-B)(A+B), and factoring perfect square trinomials using the form A^2 + 2AB + B^2 = (A+B)^2. In all, it thoroughly explains the step-by
This document provides examples and explanations of operations and concepts involving polynomial and rational expressions. It begins with examples of factoring polynomials and using the factored form to evaluate expressions. It then covers topics such as combining like terms in rational expressions, multiplying and dividing rational expressions using factoring, simplifying complex fractions, and rationalizing denominators involving radicals. The document aims to demonstrate techniques for working with polynomials and rational expressions through step-by-step examples and explanations of related concepts.
This document discusses various techniques for factoring algebraic expressions, including:
1) Factoring out common factors from terms.
2) Factoring trinomials of the form x^2 + bx + c by finding two numbers whose sum is b and product is c.
3) Using special formulas to factor differences and sums of squares and cubes.
4) Recognizing perfect squares and factoring them.
5) Factoring completely by repeated application of factoring methods.
This document discusses various techniques for factoring algebraic expressions, including:
1) Factoring out common factors from terms.
2) Factoring trinomials of the form x^2 + bx + c by finding two numbers whose sum is b and product is c.
3) Using special formulas to factor differences and sums of squares and cubes.
4) Recognizing perfect squares and factoring them.
5) Factoring completely by repeated application of factoring methods.
53 multiplication and division of rational expressionsalg1testreview
The document discusses multiplication and division of rational expressions. It presents the multiplication rule for rational expressions, which states that the product of two rational expressions is equal to the product of the tops divided by the product of the bottoms. Examples are provided to demonstrate simplifying rational expression products by factoring and canceling like terms.
4 multiplication and division of rational expressionsmath123b
The document discusses multiplication and division of rational expressions. It presents the multiplication rule for rational expressions, which states that the product of two rational expressions is equal to the product of the top expressions divided by the product of the bottom expressions. Examples are provided to demonstrate simplifying rational expression products by factoring and canceling like terms.
This document provides examples and explanations of operations involving polynomials and rational expressions. It covers factoring polynomials, evaluating polynomial expressions, adding, subtracting, multiplying and dividing rational expressions, and simplifying complex fractions and expressions with radicals. Step-by-step solutions are shown for problems such as factoring expressions, evaluating polynomials for given values, combining like rational expressions, rationalizing denominators, and more. The document demonstrates various techniques for working with polynomials and rational expressions.
The document discusses applications of factoring expressions. The main purposes of factoring an expression E into a product E=AB is to utilize properties of multiplication. The most important application of factoring is to solve polynomial equations by setting each factor equal to 0 based on the zero-product property. Examples are provided to demonstrate solving polynomial equations by factoring, setting each factor equal to 0, and extracting the solutions.
This document provides instruction on factoring polynomials. It covers several factoring methods including common monomial factoring, difference of squares, sum and difference of cubes, perfect square trinomials, and general trinomials. Examples are provided for each method. The objectives are to determine appropriate factoring methods, factor polynomials completely using various techniques, and solve problems involving polynomial factors.
3 multiplication and division of rational expressions xTzenma
The document discusses multiplication and division of rational expressions. It presents the multiplication rule for rational expressions, which is that the product of two rational expressions is equal to the product of the numerators divided by the product of the denominators. It then gives examples of simplifying products and quotients of rational expressions by factoring and canceling like terms.
Factoring is writing a polynomial as a product of two or more polynomials. The main techniques for factoring polynomials are finding the greatest common factor, factoring trinomials of the form ax^2 + bx + c, using special factoring patterns like the difference and sum of squares, and factoring polynomials with four or more terms by grouping. The goal is to factor the polynomial completely into prime factors that cannot be further factored.
This module introduces linear functions. It defines a linear function as one that can be written in the form f(x) = mx + b, where m is the slope and b is the y-intercept. It explains how to graph linear functions given two points, the x- and y-intercepts, the slope and a point, or the slope and y-intercept. The document provides examples and practice problems for students to learn how to represent linear functions in different forms, rewrite them between standard and slope-intercept form, and graph them based on given information.
13 multiplication and division of rational expressionselem-alg-sample
The document discusses multiplication and division of rational expressions. It presents the multiplication rule for rational expressions as the product of the numerators over the product of the denominators. It provides an example of simplifying a rational expression by factoring the top and bottom and canceling like terms. It then gives another example with two parts, simplifying and expanding the answers of rational expression operations.
2/27/12 Special Factoring - Sum & Difference of Two Cubesjennoga08
The document is about factoring polynomials, specifically factoring the sum and difference of cubes. It provides the formulas for factoring the sum and difference of cubes, along with examples of factoring expressions using those formulas. It also discusses factoring out the greatest common factor from polynomials.
This document provides a summary of key concepts from a college algebra textbook, including:
Rational expressions involve fractions of polynomials. The domain of an algebraic expression is the set of real numbers for which the expression is defined. Compound fractions contain fractions in the numerator, denominator, or both. Rational expressions can be simplified by factoring and canceling common factors. Adding and subtracting rational expressions requires finding a common denominator. The denominator of a fraction can be rationalized by multiplying the numerator and denominator by the conjugate radical. Common errors involve applying properties of multiplication to addition incorrectly.
The document provides examples and explanations for solving different types of equations, including:
1) Polynomial equations through factoring or the quadratic formula.
2) Rational equations by clearing denominators.
3) Radical equations by squaring both sides to remove radicals.
4) Absolute value equations by recognizing that |x-c|=r implies x=c±r.
The document also discusses solving power equations, finding zeros and domains of functions, and using properties of absolute values.
This document provides notes and practice problems about the order of operations. It explains that parentheses, exponents, multiplication, division, addition, and subtraction have a specific order in which they must be evaluated when multiple operations are present in an expression. Students are given examples of each operation and a mnemonic to remember the order of operations as PEMDAS, which stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. Guided practice problems are included for students to evaluate expressions using the proper order of operations.
This document discusses special polynomial products and factoring polynomials. It begins by covering special product rules for multiplying binomials, squaring binomials, taking the cube of a binomial, and multiplying a binomial and trinomial. It then discusses different factoring techniques, including factoring a common monomial, difference of squares, factoring trinomials, factoring perfect square trinomials, sum and difference of cubes, and factoring by grouping. The document provides examples of each type of special product and factoring technique. It aims to teach how to find special products and completely factor polynomials into prime factors.
The document discusses equations of lines. It separates lines into two cases - horizontal/vertical lines which have slope 0 or undefined slope, and their equations are y=c or x=c; and tilted lines, whose equations can be found using the point-slope formula y-y1=m(x-x1) where m is the slope and (x1,y1) is a point on the line. It provides examples of finding equations of lines given their properties like slope and intercept points.
The document discusses the concept of slope of a line. It defines slope as the ratio of the "rise" over the "run" between two points on a line. Specifically:
- The slope of a line is calculated as the change in the y-values (rise) divided by the change in the x-values (run) between two points on the line.
- This formula is easy to memorize and captures the geometric meaning of slope as the tilt of the line.
- An example problem demonstrates calculating the slope of a line between two points by finding the difference in their x- and y-values.
The document describes the rectangular coordinate system. It establishes that a coordinate system assigns positions in a plane using ordered pairs of numbers (x,y). It defines the x-axis, y-axis, and origin at their intersection. Any point is addressed by its coordinates (x,y) where x represents horizontal distance from the origin and y represents vertical distance. The four quadrants divided by the axes are also defined based on positive and negative coordinate values. Reflections of points across the axes and origin are discussed. Finally, it introduces the concept of graphing mathematical relations between x and y coordinates to represent collections of points.
The document describes the rectangular coordinate system. It defines the system as using a grid with two perpendicular axes (x and y) that intersect at the origin (0,0). Any point in the plane can be located using its coordinates (x,y), where x is the distance from the y-axis and y is the distance from the x-axis. The four quadrants (I, II, III, IV) are defined by the intersection of the positive and negative sides of the x and y axes. Examples are given of labeling points and finding coordinates on the grid.
2 the real line, inequalities and comparative phraseselem-alg-sample
The document discusses inequalities and the real number line. It explains that real numbers are associated with positions on a line, with positive numbers to the right of zero and negative numbers to the left. An inequality relates the position of two numbers on the real number line, with the number farther to the right said to be greater than the number on the left. Examples are provided of drawing intervals on the number line and solving simple inequalities algebraically. Properties of inequalities like adding the same quantity to both sides preserving the inequality sign are also outlined.
Geometry is the study of shapes, their properties and relationships. Some basic geometric shapes include lines, rays, angles, triangles, quadrilaterals, polygons, circles and three-dimensional shapes like spheres and cubes. Formulas are used to calculate properties of shapes like the area of a triangle is 1/2 * base * height, the circumference of a circle is 2 * pi * radius, and the volume of a cube is side^3.
The document discusses direct and inverse variations. It defines a direct variation as a relationship where y=kx, where k is a constant. An inverse variation is defined as a relationship where y=k/x, where k is a constant. Examples are given of translating phrases describing direct and inverse variations into mathematical equations. The document also explains how to solve word problems involving variations by using given values to find the specific constant k and exact variation equation.
17 applications of proportions and the rational equationselem-alg-sample
The document discusses rational equations word problems involving rates, distances, costs, and number of people. An example problem asks how many people (x) shared a taxi costing $20 if one person leaving causes the remaining people's cost to increase by $1 each. Setting up rational equations and solving leads to the answer that x = 5 people.
16 the multiplier method for simplifying complex fractionselem-alg-sample
The document discusses two methods for simplifying complex fractions. A complex fraction is a fraction with fractions in the numerator or denominator. The first method reduces the complex fraction to an "easy" regular division problem by combining fractions in the numerator and denominator. The second method multiplies the lowest common denominator of all terms to the numerator and denominator to simplify. An example using each method is provided.
15 proportions and the multiplier method for solving rational equationselem-alg-sample
The document discusses addition and subtraction of rational expressions. It states that rational expressions can only be added or subtracted if they have the same denominator. It provides the rule for adding or subtracting rational expressions with the same denominator. It also discusses converting rational expressions to have a common denominator so they can be added or subtracted, using the least common multiple of the denominators. Examples are provided to demonstrate converting rational expressions to equivalent forms with different specified denominators.
14 the lcm and the multiplier method for addition and subtraction of rational...elem-alg-sample
The document discusses methods for finding the least common multiple (LCM) of numbers. It defines a multiple as a number that can be divided evenly by another number. The LCM is the smallest number that is a multiple of all numbers given. Two methods are described: the searching method which tests multiples of the largest number, and the construction method which factors each number and multiplies the highest powers of common factors. Examples are provided to illustrate both methods.
Rational expressions are expressions of the form P/Q, where P and Q are polynomials. Polynomials are expressions of the form anxn + an-1xn-1 + ... + a1x1 + a0. Rational expressions can be written in either expanded or factored form. The factored form is useful for determining the domain of a rational expression, solving equations involving rational expressions, evaluating inputs, and determining the sign of outputs. The domain of a rational expression excludes values of x that make the denominator equal to 0.
10 more on factoring trinomials and factoring by formulaselem-alg-sample
The document discusses two methods for factoring trinomials of the form ax^2 + bx + c. The first method is short but not always reliable, while the second method takes more steps but always provides a definite answer. This second method, called the reversed FOIL method, involves finding four numbers that satisfy certain properties to factor the trinomial. An example is worked out step-by-step to demonstrate how to use the reversed FOIL method to factor the trinomial 3x^2 + 5x + 2.
Trinomials are polynomials of the form ax^2 + bx + c, where a, b, and c are numbers. To factor a trinomial, we write it as the product of two binomials (x + u)(x + v) where uv = c and u + v = b. For example, to factor x^2 + 5x + 6, we set uv = 6 and u + v = 5. The only possible values are u = 2 and v = 3, so x^2 + 5x + 6 = (x + 2)(x + 3). Similarly, to factor x^2 - 5x + 6, we set uv = 6 and u + v = -5,
The document discusses factoring quantities by finding common factors. It defines factoring as rewriting a quantity as a product in a nontrivial way. A quantity is prime if it cannot be written as a product other than 1 times the quantity. To factor completely means writing each factor as a product of prime numbers. Examples show finding common factors of quantities, the greatest common factor (GCF), and extracting common factors from sums and differences using the extraction law.
The document discusses methods for multiplying binomial expressions. A binomial is a two-term polynomial of the form ax + b, while a trinomial is a three-term polynomial of the form ax^2 + bx + c. The product of two binomials results in a trinomial. The FOIL method is introduced to multiply binomials, where the Front, Outer, Inner, and Last terms of each binomial are multiplied and combined. Expanding the product of a binomial and a binomial with a leading negative sign requires distributing the negative sign first before using FOIL.
The document discusses polynomial expressions. A polynomial is the sum of monomial terms, where a monomial is a number multiplied by one or more variables raised to a non-negative integer power. Examples show evaluating polynomials by substituting values for variables and calculating each monomial term separately before combining them. A term refers to each monomial in a polynomial. Terms are identified by their variable part, such as the x2-term, x-term, or constant term.
The document discusses exponents and rules for exponents. It defines exponents as representing the quantity A multiplied by itself N times, written as AN. It then presents and explains the following rules for exponents:
1) Multiplication Rule: ANAK = AN+K
2) Division Rule: AN/AK = AN-K
3) Power Rule: (AN)K = ANK
4) 0-Power Rule: A0 = 1
5) Negative Power Rule: A-K = 1/AK
It provides examples to illustrate how to apply each rule when simplifying expressions with exponents.
The document discusses solving literal equations by isolating the variable of interest on one side of the equation. It provides steps to take which include clearing fractions by multiplying both sides by the LCD, moving all terms except the variable of interest to one side of the equation, and then dividing both sides by the coefficient of the isolated variable term to solve for the variable. Examples are provided to demonstrate these steps, such as solving for x in (a + b)x = c by dividing both sides by (a + b).
The document provides examples and explanations for solving linear equations with one step. It defines a linear equation as one where both sides are linear expressions, such as 3x + 10 = 34, and not containing higher powers of x. To solve a one-step linear equation, the goal is to isolate the variable x on one side by applying the opposite operation to both sides, such as adding 3 to both sides of x - 3 = 12 to get x = 15. Worked examples are provided for solving equations of the form x ± a = b and cx = d.
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
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The simplified electron and muon model, Oscillating Spacetime: The Foundation...RitikBhardwaj56
Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
How to Setup Warehouse & Location in Odoo 17 InventoryCeline George
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Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
4. Applications of Factoring
There are many applications of the factored forms of
polynomials. Following are some of them:
1. to evaluate polynomials,
5. Applications of Factoring
There are many applications of the factored forms of
polynomials. Following are some of them:
1. to evaluate polynomials,
2. to determine the signs of the outputs,
6. Applications of Factoring
There are many applications of the factored forms of
polynomials. Following are some of them:
1. to evaluate polynomials,
2. to determine the signs of the outputs,
3. most importantly, to solve polynomial-equations.
7. Applications of Factoring
There are many applications of the factored forms of
polynomials. Following are some of them:
1. to evaluate polynomials,
2. to determine the signs of the outputs,
3. most importantly, to solve polynomial-equations.
Evaluating Polynomials
8. Applications of Factoring
There are many applications of the factored forms of
polynomials. Following are some of them:
1. to evaluate polynomials,
2. to determine the signs of the outputs,
3. most importantly, to solve polynomial-equations.
Evaluating Polynomials
Often it is easier to evaluate polynomials in the factored form.
9. Applications of Factoring
There are many applications of the factored forms of
polynomials. Following are some of them:
1. to evaluate polynomials,
2. to determine the signs of the outputs,
3. most importantly, to solve polynomial-equations.
Evaluating Polynomials
Example A. Evaluate x2 – 2x – 3 if x = 7
a. without factoring. b. by factoring it first.
Often it is easier to evaluate polynomials in the factored form.
10. Applications of Factoring
There are many applications of the factored forms of
polynomials. Following are some of them:
1. to evaluate polynomials,
2. to determine the signs of the outputs,
3. most importantly, to solve polynomial-equations.
Evaluating Polynomials
Example A. Evaluate x2 – 2x – 3 if x = 7
a. without factoring. b. by factoring it first.
We get
72 – 2(7) – 3
Often it is easier to evaluate polynomials in the factored form.
11. Applications of Factoring
There are many applications of the factored forms of
polynomials. Following are some of them:
1. to evaluate polynomials,
2. to determine the signs of the outputs,
3. most importantly, to solve polynomial-equations.
Evaluating Polynomials
Example A. Evaluate x2 – 2x – 3 if x = 7
a. without factoring. b. by factoring it first.
We get
72 – 2(7) – 3
= 49 – 14 – 3
Often it is easier to evaluate polynomials in the factored form.
12. Applications of Factoring
There are many applications of the factored forms of
polynomials. Following are some of them:
1. to evaluate polynomials,
2. to determine the signs of the outputs,
3. most importantly, to solve polynomial-equations.
Evaluating Polynomials
Example A. Evaluate x2 – 2x – 3 if x = 7
a. without factoring. b. by factoring it first.
We get
72 – 2(7) – 3
= 49 – 14 – 3
= 32
Often it is easier to evaluate polynomials in the factored form.
13. Applications of Factoring
There are many applications of the factored forms of
polynomials. Following are some of them:
1. to evaluate polynomials,
2. to determine the signs of the outputs,
3. most importantly, to solve polynomial-equations.
Evaluating Polynomials
Example A. Evaluate x2 – 2x – 3 if x = 7
a. without factoring. b. by factoring it first.
We get
72 – 2(7) – 3
= 49 – 14 – 3
= 32
x2 – 2x – 3 = (x – 3)(x+1)
Often it is easier to evaluate polynomials in the factored form.
14. Applications of Factoring
There are many applications of the factored forms of
polynomials. Following are some of them:
1. to evaluate polynomials,
2. to determine the signs of the outputs,
3. most importantly, to solve polynomial-equations.
Evaluating Polynomials
Example A. Evaluate x2 – 2x – 3 if x = 7
a. without factoring. b. by factoring it first.
We get
72 – 2(7) – 3
= 49 – 14 – 3
= 32
x2 – 2x – 3 = (x – 3)(x+1)
We get
(7 – 3)(7 + 1)
Often it is easier to evaluate polynomials in the factored form.
15. Applications of Factoring
There are many applications of the factored forms of
polynomials. Following are some of them:
1. to evaluate polynomials,
2. to determine the signs of the outputs,
3. most importantly, to solve polynomial-equations.
Evaluating Polynomials
Example A. Evaluate x2 – 2x – 3 if x = 7
a. without factoring. b. by factoring it first.
We get
72 – 2(7) – 3
= 49 – 14 – 3
= 32
x2 – 2x – 3 = (x – 3)(x+1)
We get
(7 – 3)(7 + 1)
= 4(8)
= 32
Often it is easier to evaluate polynomials in the factored form.
16. Applications of Factoring
There are many applications of the factored forms of
polynomials. Following are some of them:
1. to evaluate polynomials,
2. to determine the signs of the outputs,
3. most importantly, to solve polynomial-equations.
Evaluating Polynomials
Example A. Evaluate x2 – 2x – 3 if x = 7
a. without factoring. b. by factoring it first.
We get
72 – 2(7) – 3
= 49 – 14 – 3
= 32
x2 – 2x – 3 = (x – 3)(x+1)
We get
(7 – 3)(7 + 1)
= 4(8)
= 32
Often it is easier to evaluate polynomials in the factored form.
17. Example B. Evaluate 2x3 – 5x2 + 2x
for x = -2, -1, 3 by factoring it first.
Applications of Factoring
18. Example B. Evaluate 2x3 – 5x2 + 2x
for x = -2, -1, 3 by factoring it first.
2x3 – 5x2 + 2x = x(2x2 – 5x + 2)
Applications of Factoring
19. Example B. Evaluate 2x3 – 5x2 + 2x
for x = -2, -1, 3 by factoring it first.
2x3 – 5x2 + 2x = x(2x2 – 5x + 2)
= x(2x – 1)(x – 2)
Applications of Factoring
20. Example B. Evaluate 2x3 – 5x2 + 2x
for x = -2, -1, 3 by factoring it first.
2x3 – 5x2 + 2x = x(2x2 – 5x + 2)
= x(2x – 1)(x – 2)
For x = -2:
(-2)[2(-2) – 1] [(-2) – 2]
Applications of Factoring
21. Example B. Evaluate 2x3 – 5x2 + 2x
for x = -2, -1, 3 by factoring it first.
2x3 – 5x2 + 2x = x(2x2 – 5x + 2)
= x(2x – 1)(x – 2)
For x = -2:
(-2)[2(-2) – 1] [(-2) – 2] = -2 [-5] [-4]
Applications of Factoring
22. Example B. Evaluate 2x3 – 5x2 + 2x
for x = -2, -1, 3 by factoring it first.
2x3 – 5x2 + 2x = x(2x2 – 5x + 2)
= x(2x – 1)(x – 2)
For x = -2:
(-2)[2(-2) – 1] [(-2) – 2] = -2 [-5] [-4] = -40
Applications of Factoring
23. Example B. Evaluate 2x3 – 5x2 + 2x
for x = -2, -1, 3 by factoring it first.
2x3 – 5x2 + 2x = x(2x2 – 5x + 2)
= x(2x – 1)(x – 2)
For x = -2:
(-2)[2(-2) – 1] [(-2) – 2] = -2 [-5] [-4] = -40
For x = -1:
(-1)[2(-1) – 1] [(-1) – 2]
Applications of Factoring
24. Example B. Evaluate 2x3 – 5x2 + 2x
for x = -2, -1, 3 by factoring it first.
2x3 – 5x2 + 2x = x(2x2 – 5x + 2)
= x(2x – 1)(x – 2)
For x = -2:
(-2)[2(-2) – 1] [(-2) – 2] = -2 [-5] [-4] = -40
For x = -1:
(-1)[2(-1) – 1] [(-1) – 2] = -1 [-3] [-3]
Applications of Factoring
25. Example B. Evaluate 2x3 – 5x2 + 2x
for x = -2, -1, 3 by factoring it first.
2x3 – 5x2 + 2x = x(2x2 – 5x + 2)
= x(2x – 1)(x – 2)
For x = -2:
(-2)[2(-2) – 1] [(-2) – 2] = -2 [-5] [-4] = -40
For x = -1:
(-1)[2(-1) – 1] [(-1) – 2] = -1 [-3] [-3] = -9
Applications of Factoring
26. Example B. Evaluate 2x3 – 5x2 + 2x
for x = -2, -1, 3 by factoring it first.
2x3 – 5x2 + 2x = x(2x2 – 5x + 2)
= x(2x – 1)(x – 2)
For x = -2:
(-2)[2(-2) – 1] [(-2) – 2] = -2 [-5] [-4] = -40
For x = -1:
(-1)[2(-1) – 1] [(-1) – 2] = -1 [-3] [-3] = -9
For x = 3:
3 [2(3) – 1] [(3) – 2]
Applications of Factoring
27. Example B. Evaluate 2x3 – 5x2 + 2x
for x = -2, -1, 3 by factoring it first.
2x3 – 5x2 + 2x = x(2x2 – 5x + 2)
= x(2x – 1)(x – 2)
For x = -2:
(-2)[2(-2) – 1] [(-2) – 2] = -2 [-5] [-4] = -40
For x = -1:
(-1)[2(-1) – 1] [(-1) – 2] = -1 [-3] [-3] = -9
For x = 3:
3 [2(3) – 1] [(3) – 2] = 3 [5] [1] = 15
Applications of Factoring
28. Example B. Evaluate 2x3 – 5x2 + 2x
for x = -2, -1, 3 by factoring it first.
2x3 – 5x2 + 2x = x(2x2 – 5x + 2)
= x(2x – 1)(x – 2)
For x = -2:
(-2)[2(-2) – 1] [(-2) – 2] = -2 [-5] [-4] = -40
For x = -1:
(-1)[2(-1) – 1] [(-1) – 2] = -1 [-3] [-3] = -9
For x = 3:
3 [2(3) – 1] [(3) – 2] = 3 [5] [1] = 15
Applications of Factoring
29. Example B. Evaluate 2x3 – 5x2 + 2x
for x = -2, -1, 3 by factoring it first.
2x3 – 5x2 + 2x = x(2x2 – 5x + 2)
= x(2x – 1)(x – 2)
For x = -2:
(-2)[2(-2) – 1] [(-2) – 2] = -2 [-5] [-4] = -40
For x = -1:
(-1)[2(-1) – 1] [(-1) – 2] = -1 [-3] [-3] = -9
For x = 3:
3 [2(3) – 1] [(3) – 2] = 3 [5] [1] = 15
Applications of Factoring
Your turn: Double check these answers via the expanded form.
30. Example B. Evaluate 2x3 – 5x2 + 2x
for x = -2, -1, 3 by factoring it first.
2x3 – 5x2 + 2x = x(2x2 – 5x + 2)
= x(2x – 1)(x – 2)
For x = -2:
(-2)[2(-2) – 1] [(-2) – 2] = -2 [-5] [-4] = -40
For x = -1:
(-1)[2(-1) – 1] [(-1) – 2] = -1 [-3] [-3] = -9
For x = 3:
3 [2(3) – 1] [(3) – 2] = 3 [5] [1] = 15
Applications of Factoring
Determine the signs of the outputs
Your turn: Double check these answers via the expanded form.
31. Example B. Evaluate 2x3 – 5x2 + 2x
for x = -2, -1, 3 by factoring it first.
2x3 – 5x2 + 2x = x(2x2 – 5x + 2)
= x(2x – 1)(x – 2)
For x = -2:
(-2)[2(-2) – 1] [(-2) – 2] = -2 [-5] [-4] = -40
For x = -1:
(-1)[2(-1) – 1] [(-1) – 2] = -1 [-3] [-3] = -9
For x = 3:
3 [2(3) – 1] [(3) – 2] = 3 [5] [1] = 15
Applications of Factoring
Determine the signs of the outputs
Often we only want to know the sign of the output, i.e.
whether the output is positive or negative.
Your turn: Double check these answers via the expanded form.
32. Example B. Evaluate 2x3 – 5x2 + 2x
for x = -2, -1, 3 by factoring it first.
2x3 – 5x2 + 2x = x(2x2 – 5x + 2)
= x(2x – 1)(x – 2)
For x = -2:
(-2)[2(-2) – 1] [(-2) – 2] = -2 [-5] [-4] = -40
For x = -1:
(-1)[2(-1) – 1] [(-1) – 2] = -1 [-3] [-3] = -9
For x = 3:
3 [2(3) – 1] [(3) – 2] = 3 [5] [1] = 15
Applications of Factoring
Determine the signs of the outputs
Often we only want to know the sign of the output, i.e.
whether the output is positive or negative. It is easy to do this
using the factored form.
Your turn: Double check these answers via the expanded form.
33. Example C. Determine the outcome is + or – for x2 – 2x – 3
if x = -3/2, -1/2.
Applications of Factoring
34. Example C. Determine the outcome is + or – for x2 – 2x – 3
if x = -3/2, -1/2.
Factor x2 – 2x – 3 = (x – 3)(x + 1)
Applications of Factoring
35. Example C. Determine the outcome is + or – for x2 – 2x – 3
if x = -3/2, -1/2.
Factor x2 – 2x – 3 = (x – 3)(x + 1)
Hence for x = -3/2:
(-3/2 – 3)(-3/2 + 1)
Applications of Factoring
36. Example C. Determine the outcome is + or – for x2 – 2x – 3
if x = -3/2, -1/2.
Factor x2 – 2x – 3 = (x – 3)(x + 1)
Hence for x = -3/2:
(-3/2 – 3)(-3/2 + 1) is (–)(–) = + .
Applications of Factoring
37. Example C. Determine the outcome is + or – for x2 – 2x – 3
if x = -3/2, -1/2.
Factor x2 – 2x – 3 = (x – 3)(x + 1)
Hence for x = -3/2:
(-3/2 – 3)(-3/2 + 1) is (–)(–) = + . So the outcome is positive.
Applications of Factoring
38. Example C. Determine the outcome is + or – for x2 – 2x – 3
if x = -3/2, -1/2.
Factor x2 – 2x – 3 = (x – 3)(x + 1)
Hence for x = -3/2:
(-3/2 – 3)(-3/2 + 1) is (–)(–) = + . So the outcome is positive.
And for x = -1/2:
(-1/2 – 3)(-1/2 + 1)
Applications of Factoring
39. Example C. Determine the outcome is + or – for x2 – 2x – 3
if x = -3/2, -1/2.
Factor x2 – 2x – 3 = (x – 3)(x + 1)
Hence for x = -3/2:
(-3/2 – 3)(-3/2 + 1) is (–)(–) = + . So the outcome is positive.
And for x = -1/2:
(-1/2 – 3)(-1/2 + 1) is (–)(+) = – .
Applications of Factoring
40. Example C. Determine the outcome is + or – for x2 – 2x – 3
if x = -3/2, -1/2.
Factor x2 – 2x – 3 = (x – 3)(x + 1)
Hence for x = -3/2:
(-3/2 – 3)(-3/2 + 1) is (–)(–) = + . So the outcome is positive.
And for x = -1/2:
(-1/2 – 3)(-1/2 + 1) is (–)(+) = – . So the outcome is negative.
Applications of Factoring
41. Example C. Determine the outcome is + or – for x2 – 2x – 3
if x = -3/2, -1/2.
Factor x2 – 2x – 3 = (x – 3)(x + 1)
Hence for x = -3/2:
(-3/2 – 3)(-3/2 + 1) is (–)(–) = + . So the outcome is positive.
And for x = -1/2:
(-1/2 – 3)(-1/2 + 1) is (–)(+) = – . So the outcome is negative.
Applications of Factoring
Solving Equations
42. Example C. Determine the outcome is + or – for x2 – 2x – 3
if x = -3/2, -1/2.
Factor x2 – 2x – 3 = (x – 3)(x + 1)
Hence for x = -3/2:
(-3/2 – 3)(-3/2 + 1) is (–)(–) = + . So the outcome is positive.
And for x = -1/2:
(-1/2 – 3)(-1/2 + 1) is (–)(+) = – . So the outcome is negative.
Applications of Factoring
Solving Equations
The most important application for factoring is to solve
polynomial equations.
43. Example C. Determine the outcome is + or – for x2 – 2x – 3
if x = -3/2, -1/2.
Factor x2 – 2x – 3 = (x – 3)(x + 1)
Hence for x = -3/2:
(-3/2 – 3)(-3/2 + 1) is (–)(–) = + . So the outcome is positive.
And for x = -1/2:
(-1/2 – 3)(-1/2 + 1) is (–)(+) = – . So the outcome is negative.
Applications of Factoring
Solving Equations
The most important application for factoring is to solve
polynomial equations. These are equations of the form
polynomial = polynomial
44. Example C. Determine the outcome is + or – for x2 – 2x – 3
if x = -3/2, -1/2.
Factor x2 – 2x – 3 = (x – 3)(x + 1)
Hence for x = -3/2:
(-3/2 – 3)(-3/2 + 1) is (–)(–) = + . So the outcome is positive.
And for x = -1/2:
(-1/2 – 3)(-1/2 + 1) is (–)(+) = – . So the outcome is negative.
Applications of Factoring
Solving Equations
The most important application for factoring is to solve
polynomial equations. These are equations of the form
polynomial = polynomial
To solve these equations, we use the following obvious fact.
45. Example C. Determine the outcome is + or – for x2 – 2x – 3
if x = -3/2, -1/2.
Factor x2 – 2x – 3 = (x – 3)(x + 1)
Hence for x = -3/2:
(-3/2 – 3)(-3/2 + 1) is (–)(–) = + . So the outcome is positive.
And for x = -1/2:
(-1/2 – 3)(-1/2 + 1) is (–)(+) = – . So the outcome is negative.
Applications of Factoring
Solving Equations
The most important application for factoring is to solve
polynomial equations. These are equations of the form
polynomial = polynomial
To solve these equations, we use the following obvious fact.
Fact: If A*B = 0,
then either A = 0 or B = 0
46. Example C. Determine the outcome is + or – for x2 – 2x – 3
if x = -3/2, -1/2.
Factor x2 – 2x – 3 = (x – 3)(x + 1)
Hence for x = -3/2:
(-3/2 – 3)(-3/2 + 1) is (–)(–) = + . So the outcome is positive.
And for x = -1/2:
(-1/2 – 3)(-1/2 + 1) is (–)(+) = – . So the outcome is negative.
Applications of Factoring
Solving Equations
The most important application for factoring is to solve
polynomial equations. These are equations of the form
polynomial = polynomial
To solve these equations, we use the following obvious fact.
Fact: If A*B = 0,
then either A = 0 or B = 0
For example, if 3x = 0, then x must be equal to 0.
48. Example D.
a. If 3(x – 2) = 0, then (x – 2) = 0,
Applications of Factoring
49. Example D.
a. If 3(x – 2) = 0, then (x – 2) = 0, so x must be 2.
Applications of Factoring
50. Example D.
a. If 3(x – 2) = 0, then (x – 2) = 0, so x must be 2.
Applications of Factoring
b. If (x + 1)(x – 2) = 0,
51. Example D.
a. If 3(x – 2) = 0, then (x – 2) = 0, so x must be 2.
Applications of Factoring
b. If (x + 1)(x – 2) = 0,
then either (x + 1) = 0 or (x – 2) = 0,
52. Example D.
a. If 3(x – 2) = 0, then (x – 2) = 0, so x must be 2.
Applications of Factoring
b. If (x + 1)(x – 2) = 0,
then either (x + 1) = 0 or (x – 2) = 0,
x = –1
53. Example D.
a. If 3(x – 2) = 0, then (x – 2) = 0, so x must be 2.
Applications of Factoring
b. If (x + 1)(x – 2) = 0,
then either (x + 1) = 0 or (x – 2) = 0,
x = –1 or x = 2
54. Example D.
a. If 3(x – 2) = 0, then (x – 2) = 0, so x must be 2.
Applications of Factoring
To solve polynomial equation,
b. If (x + 1)(x – 2) = 0,
then either (x + 1) = 0 or (x – 2) = 0,
x = –1 or x = 2
55. Example D.
a. If 3(x – 2) = 0, then (x – 2) = 0, so x must be 2.
Applications of Factoring
To solve polynomial equation,
1. set one side of the equation to be 0, move all the terms to
the other side.
b. If (x + 1)(x – 2) = 0,
then either (x + 1) = 0 or (x – 2) = 0,
x = –1 or x = 2
56. Example D.
a. If 3(x – 2) = 0, then (x – 2) = 0, so x must be 2.
Applications of Factoring
To solve polynomial equation,
1. set one side of the equation to be 0, move all the terms to
the other side.
2. factor the polynomial,
b. If (x + 1)(x – 2) = 0,
then either (x + 1) = 0 or (x – 2) = 0,
x = –1 or x = 2
57. Example D.
a. If 3(x – 2) = 0, then (x – 2) = 0, so x must be 2.
Applications of Factoring
To solve polynomial equation,
1. set one side of the equation to be 0, move all the terms to
the other side.
2. factor the polynomial,
3. get the answers.
b. If (x + 1)(x – 2) = 0,
then either (x + 1) = 0 or (x – 2) = 0,
x = –1 or x = 2
58. Example D.
a. If 3(x – 2) = 0, then (x – 2) = 0, so x must be 2.
Applications of Factoring
To solve polynomial equation,
1. set one side of the equation to be 0, move all the terms to
the other side.
2. factor the polynomial,
3. get the answers.
b. If (x + 1)(x – 2) = 0,
then either (x + 1) = 0 or (x – 2) = 0,
x = –1 or x = 2
Example E. Solve for x
a. x2 – 2x – 3 = 0
59. Example D.
a. If 3(x – 2) = 0, then (x – 2) = 0, so x must be 2.
Applications of Factoring
To solve polynomial equation,
1. set one side of the equation to be 0, move all the terms to
the other side.
2. factor the polynomial,
3. get the answers.
b. If (x + 1)(x – 2) = 0,
then either (x + 1) = 0 or (x – 2) = 0,
x = –1 or x = 2
Example E. Solve for x
a. x2 – 2x – 3 = 0 Factor
(x – 3)(x + 1) = 0
60. Example D.
a. If 3(x – 2) = 0, then (x – 2) = 0, so x must be 2.
Applications of Factoring
To solve polynomial equation,
1. set one side of the equation to be 0, move all the terms to
the other side.
2. factor the polynomial,
3. get the answers.
b. If (x + 1)(x – 2) = 0,
then either (x + 1) = 0 or (x – 2) = 0,
x = –1 or x = 2
Example E. Solve for x
a. x2 – 2x – 3 = 0 Factor
(x – 3)(x + 1) = 0
There are two linear
x–factors. We may extract
one answer from each.
61. Example D.
a. If 3(x – 2) = 0, then (x – 2) = 0, so x must be 2.
Applications of Factoring
To solve polynomial equation,
1. set one side of the equation to be 0, move all the terms to
the other side.
2. factor the polynomial,
3. get the answers.
b. If (x + 1)(x – 2) = 0,
then either (x + 1) = 0 or (x – 2) = 0,
x = –1 or x = 2
Example E. Solve for x
a. x2 – 2x – 3 = 0 Factor
(x – 3)(x + 1) = 0
Hence x – 3 = 0 or x + 1 = 0
There are two linear
x–factors. We may extract
one answer from each.
62. Example D.
a. If 3(x – 2) = 0, then (x – 2) = 0, so x must be 2.
Applications of Factoring
To solve polynomial equation,
1. set one side of the equation to be 0, move all the terms to
the other side.
2. factor the polynomial,
3. get the answers.
b. If (x + 1)(x – 2) = 0,
then either (x + 1) = 0 or (x – 2) = 0,
x = –1 or x = 2
Example E. Solve for x
a. x2 – 2x – 3 = 0 Factor
(x – 3)(x + 1) = 0
Hence x – 3 = 0 or x + 1 = 0
x = 3
There are two linear
x–factors. We may extract
one answer from each.
63. Example D.
a. If 3(x – 2) = 0, then (x – 2) = 0, so x must be 2.
Applications of Factoring
To solve polynomial equation,
1. set one side of the equation to be 0, move all the terms to
the other side.
2. factor the polynomial,
3. get the answers.
b. If (x + 1)(x – 2) = 0,
then either (x + 1) = 0 or (x – 2) = 0,
x = –1 or x = 2
Example E. Solve for x
a. x2 – 2x – 3 = 0 Factor
(x – 3)(x + 1) = 0
Hence x – 3 = 0 or x + 1 = 0
x = 3 or x = -1
There are two linear
x–factors. We may extract
one answer from each.
64. b. 2x(x + 1) = 4x + 3(1 – x)
Applications of Factoring
67. b. 2x(x + 1) = 4x + 3(1 – x) Expand
2x2 + 2x = 4x + 3 – 3x
2x2 + 2x = x + 3 Set one side 0
2x2 + 2x – x – 3 = 0
Applications of Factoring
68. b. 2x(x + 1) = 4x + 3(1 – x) Expand
2x2 + 2x = 4x + 3 – 3x
2x2 + 2x = x + 3 Set one side 0
2x2 + 2x – x – 3 = 0
2x2 + x – 3 = 0
Applications of Factoring
69. b. 2x(x + 1) = 4x + 3(1 – x) Expand
2x2 + 2x = 4x + 3 – 3x
2x2 + 2x = x + 3 Set one side 0
2x2 + 2x – x – 3 = 0
2x2 + x – 3 = 0 Factor
(2x + 3)(x – 1) = 0
Applications of Factoring
70. b. 2x(x + 1) = 4x + 3(1 – x) Expand
2x2 + 2x = 4x + 3 – 3x
2x2 + 2x = x + 3 Set one side 0
2x2 + 2x – x – 3 = 0
2x2 + x – 3 = 0 Factor
(2x + 3)(x – 1) = 0 Get the answers
2x + 3 = 0
Applications of Factoring
or x – 1 = 0
71. b. 2x(x + 1) = 4x + 3(1 – x) Expand
2x2 + 2x = 4x + 3 – 3x
2x2 + 2x = x + 3 Set one side 0
2x2 + 2x – x – 3 = 0
2x2 + x – 3 = 0 Factor
(2x + 3)(x – 1) = 0 Get the answers
2x + 3 = 0
2x = -3
Applications of Factoring
or x – 1 = 0
72. b. 2x(x + 1) = 4x + 3(1 – x) Expand
2x2 + 2x = 4x + 3 – 3x
2x2 + 2x = x + 3 Set one side 0
2x2 + 2x – x – 3 = 0
2x2 + x – 3 = 0 Factor
(2x + 3)(x – 1) = 0 Get the answers
2x + 3 = 0
2x = -3
x = -3/2
Applications of Factoring
or x – 1 = 0
73. b. 2x(x + 1) = 4x + 3(1 – x) Expand
2x2 + 2x = 4x + 3 – 3x
2x2 + 2x = x + 3 Set one side 0
2x2 + 2x – x – 3 = 0
2x2 + x – 3 = 0 Factor
(2x + 3)(x – 1) = 0 Get the answers
2x + 3 = 0
2x = -3
x = -3/2
Applications of Factoring
or x – 1 = 0
x = 1
74. b. 2x(x + 1) = 4x + 3(1 – x) Expand
2x2 + 2x = 4x + 3 – 3x
2x2 + 2x = x + 3 Set one side 0
2x2 + 2x – x – 3 = 0
2x2 + x – 3 = 0 Factor
(2x + 3)(x – 1) = 0 Get the answers
2x + 3 = 0
2x = -3
x = -3/2
Applications of Factoring
or x – 1 = 0
x = 1
c. 8x(x2 – 1) = 10x
75. b. 2x(x + 1) = 4x + 3(1 – x) Expand
2x2 + 2x = 4x + 3 – 3x
2x2 + 2x = x + 3 Set one side 0
2x2 + 2x – x – 3 = 0
2x2 + x – 3 = 0 Factor
(2x + 3)(x – 1) = 0 Get the answers
2x + 3 = 0
2x = -3
x = -3/2
Applications of Factoring
or x – 1 = 0
x = 1
c. 8x(x2 – 1) = 10x Expand
8x3 – 8x = 10x
76. b. 2x(x + 1) = 4x + 3(1 – x) Expand
2x2 + 2x = 4x + 3 – 3x
2x2 + 2x = x + 3 Set one side 0
2x2 + 2x – x – 3 = 0
2x2 + x – 3 = 0 Factor
(2x + 3)(x – 1) = 0 Get the answers
2x + 3 = 0
2x = -3
x = -3/2
Applications of Factoring
or x – 1 = 0
x = 1
c. 8x(x2 – 1) = 10x Expand
8x3 – 8x = 10x Set one side 0
8x3 – 8x – 10x = 0
77. b. 2x(x + 1) = 4x + 3(1 – x) Expand
2x2 + 2x = 4x + 3 – 3x
2x2 + 2x = x + 3 Set one side 0
2x2 + 2x – x – 3 = 0
2x2 + x – 3 = 0 Factor
(2x + 3)(x – 1) = 0 Get the answers
2x + 3 = 0
2x = -3
x = -3/2
Applications of Factoring
or x – 1 = 0
x = 1
c. 8x(x2 – 1) = 10x Expand
8x3 – 8x = 10x Set one side 0
8x3 – 8x – 10x = 0
8x3 – 18x = 0
78. b. 2x(x + 1) = 4x + 3(1 – x) Expand
2x2 + 2x = 4x + 3 – 3x
2x2 + 2x = x + 3 Set one side 0
2x2 + 2x – x – 3 = 0
2x2 + x – 3 = 0 Factor
(2x + 3)(x – 1) = 0 Get the answers
2x + 3 = 0
2x = -3
x = -3/2
Applications of Factoring
or x – 1 = 0
x = 1
c. 8x(x2 – 1) = 10x Expand
8x3 – 8x = 10x Set one side 0
8x3 – 8x – 10x = 0
8x3 – 18x = 0 Factor
2x(2x + 3)(2x – 3) = 0
79. b. 2x(x + 1) = 4x + 3(1 – x) Expand
2x2 + 2x = 4x + 3 – 3x
2x2 + 2x = x + 3 Set one side 0
2x2 + 2x – x – 3 = 0
2x2 + x – 3 = 0 Factor
(2x + 3)(x – 1) = 0 Get the answers
2x + 3 = 0
2x = -3
x = -3/2
Applications of Factoring
or x – 1 = 0
x = 1
c. 8x(x2 – 1) = 10x Expand
8x3 – 8x = 10x Set one side 0
8x3 – 8x – 10x = 0
8x3 – 18x = 0 Factor
2x(2x + 3)(2x – 3) = 0
There are three linear
x–factors. We may extract
one answer from each.
80. b. 2x(x + 1) = 4x + 3(1 – x) Expand
2x2 + 2x = 4x + 3 – 3x
2x2 + 2x = x + 3 Set one side 0
2x2 + 2x – x – 3 = 0
2x2 + x – 3 = 0 Factor
(2x + 3)(x – 1) = 0 Get the answers
2x + 3 = 0
2x = -3
x = -3/2
Applications of Factoring
or x – 1 = 0
x = 1
c. 8x(x2 – 1) = 10x Expand
8x3 – 8x = 10x Set one side 0
8x3 – 8x – 10x = 0
8x3 – 18x = 0 Factor
2x(2x + 3)(2x – 3) = 0
x = 0
There are three linear
x–factors. We may extract
one answer from each.
81. b. 2x(x + 1) = 4x + 3(1 – x) Expand
2x2 + 2x = 4x + 3 – 3x
2x2 + 2x = x + 3 Set one side 0
2x2 + 2x – x – 3 = 0
2x2 + x – 3 = 0 Factor
(2x + 3)(x – 1) = 0 Get the answers
2x + 3 = 0
2x = -3
x = -3/2
Applications of Factoring
or x – 1 = 0
x = 1
c. 8x(x2 – 1) = 10x Expand
8x3 – 8x = 10x Set one side 0
8x3 – 8x – 10x = 0
8x3 – 18x = 0 Factor
2x(2x + 3)(2x – 3) = 0
x = 0 or 2x + 3 = 0
There are three linear
x–factors. We may extract
one answer from each.
82. b. 2x(x + 1) = 4x + 3(1 – x) Expand
2x2 + 2x = 4x + 3 – 3x
2x2 + 2x = x + 3 Set one side 0
2x2 + 2x – x – 3 = 0
2x2 + x – 3 = 0 Factor
(2x + 3)(x – 1) = 0 Get the answers
2x + 3 = 0
2x = -3
x = -3/2
Applications of Factoring
or x – 1 = 0
x = 1
c. 8x(x2 – 1) = 10x Expand
8x3 – 8x = 10x Set one side 0
8x3 – 8x – 10x = 0
8x3 – 18x = 0 Factor
2x(2x + 3)(2x – 3) = 0
x = 0 or 2x + 3 = 0 or 2x – 3 = 0
There are three linear
x–factors. We may extract
one answer from each.
83. b. 2x(x + 1) = 4x + 3(1 – x) Expand
2x2 + 2x = 4x + 3 – 3x
2x2 + 2x = x + 3 Set one side 0
2x2 + 2x – x – 3 = 0
2x2 + x – 3 = 0 Factor
(2x + 3)(x – 1) = 0 Get the answers
2x + 3 = 0
2x = -3
x = -3/2
Applications of Factoring
or x – 1 = 0
x = 1
c. 8x(x2 – 1) = 10x Expand
8x3 – 8x = 10x Set one side 0
8x3 – 8x – 10x = 0
8x3 – 18x = 0 Factor
2x(2x + 3)(2x – 3) = 0
x = 0 or 2x + 3 = 0 or 2x – 3 = 0
2x = -3
There are three linear
x–factors. We may extract
one answer from each.
84. b. 2x(x + 1) = 4x + 3(1 – x) Expand
2x2 + 2x = 4x + 3 – 3x
2x2 + 2x = x + 3 Set one side 0
2x2 + 2x – x – 3 = 0
2x2 + x – 3 = 0 Factor
(2x + 3)(x – 1) = 0 Get the answers
2x + 3 = 0
2x = -3
x = -3/2
Applications of Factoring
or x – 1 = 0
x = 1
c. 8x(x2 – 1) = 10x Expand
8x3 – 8x = 10x Set one side 0
8x3 – 8x – 10x = 0
8x3 – 18x = 0 Factor
2x(2x + 3)(2x – 3) = 0
x = 0 or 2x + 3 = 0 or 2x – 3 = 0
2x = -3
x = -3/2
There are three linear
x–factors. We may extract
one answer from each.
85. b. 2x(x + 1) = 4x + 3(1 – x) Expand
2x2 + 2x = 4x + 3 – 3x
2x2 + 2x = x + 3 Set one side 0
2x2 + 2x – x – 3 = 0
2x2 + x – 3 = 0 Factor
(2x + 3)(x – 1) = 0 Get the answers
2x + 3 = 0
2x = -3
x = -3/2
Applications of Factoring
or x – 1 = 0
x = 1
c. 8x(x2 – 1) = 10x Expand
8x3 – 8x = 10x Set one side 0
8x3 – 8x – 10x = 0
8x3 – 18x = 0 Factor
2x(2x + 3)(2x – 3) = 0
x = 0 or 2x + 3 = 0 or 2x – 3 = 0
2x = -3 2x = 3
x = -3/2
There are three linear
x–factors. We may extract
one answer from each.
86. b. 2x(x + 1) = 4x + 3(1 – x) Expand
2x2 + 2x = 4x + 3 – 3x
2x2 + 2x = x + 3 Set one side 0
2x2 + 2x – x – 3 = 0
2x2 + x – 3 = 0 Factor
(2x + 3)(x – 1) = 0 Get the answers
2x + 3 = 0
2x = -3
x = -3/2
Applications of Factoring
or x – 1 = 0
x = 1
c. 8x(x2 – 1) = 10x Expand
8x3 – 8x = 10x Set one side 0
8x3 – 8x – 10x = 0
8x3 – 18x = 0 Factor
2x(2x + 3)(2x – 3) = 0
x = 0 or 2x + 3 = 0 or 2x – 3 = 0
2x = -3 2x = 3
x = -3/2 x = 3/2
There are three linear
x–factors. We may extract
one answer from each.
87. Exercise A. Use the factored form to evaluate the following
expressions with the given input values.
Applications of Factoring
1. x2 – 3x – 4, x = –2, 3, 5 2. x2 – 2x – 15, x = –1, 4, 7
3. x2 – 2x – 1, x = ½ ,–2, –½ 4. x3 – 2x2, x = –2, 2, 4
5. x3 – 4x2 – 5x, x = –4, 2, 6 6. 2x3 – 3x2 + x, x = –3, 3, 5
B. Determine if the output is positive or negative using the
factored form.
7. x2 – 3x – 4, x = –2½, –2/3, 2½, 5¼
8. –x2 + 2x + 8, x = –2½, –2/3, 2½, 5¼
9. x3 – 2x2 – 8x, x = –4½, –3/4, ¼, 6¼,
11. 4x2 – x3, x = –1.22, 0.87, 3.22, 4.01
12. 18x – 2x3, x = –4.90, –2.19, 1.53, 3.01
10. 2x3 – 3x2 – 2x, x = –2½, –3/4, ¼, 3¼,