The document summarizes Chapter 13 on factoring polynomials. It covers the greatest common factor, factoring trinomials of the forms x^2 + bx + c, ax^2 + bx + c, and x^2 + bx + c by grouping. It also discusses factoring perfect square trinomials, difference of two squares, and solving quadratic equations by factoring. Examples are provided to demonstrate each technique.
This document outlines key concepts and examples for factoring polynomials. It covers factoring trinomials of the forms x^2 + bx + c, ax^2 + bx + c, and x^2 + bx + c by grouping. Examples are provided to demonstrate finding the greatest common factor of terms, factoring trinomials by finding two numbers whose product and sum meet the given criteria, and checking factoring results using FOIL multiplication. Sections include the greatest common factor, factoring trinomials of the forms x^2 + bx + c and ax^2 + bx + c.
This document outlines key concepts and examples for factoring polynomials. It discusses factoring trinomials of the forms x^2 + bx + c, ax^2 + bx + c, and x^2 + bx + c by grouping. Examples are provided to demonstrate finding the greatest common factor of terms, factoring trinomials by finding two numbers whose product and sum meet the given criteria, and checking factoring results using FOIL. Sections cover the greatest common factor, factoring trinomials of different forms, and solving quadratic equations by factoring.
This document provides an overview of Chapter 13 from a developmental mathematics textbook. It covers factoring polynomials, including:
- Finding the greatest common factor of polynomials
- Factoring trinomials of the form x^2 + bx + c and ax^2 + bx + c
- Checking factored results by multiplying the factors
The chapter sections are described and examples are provided for each type of factoring technique.
This document provides instructions on factoring polynomials of various forms:
1) It explains how to factor polynomials by finding the greatest common factor (GCF).
2) It describes how to factor trinomials of the form x2 + bx + c by finding two numbers whose sum is b and product is c.
3) It shows how to factor trinomials of the form ax2 + bx + c by finding factors of a, c whose products and sums satisfy the polynomial.
This document discusses various methods for factoring polynomials, including:
1) Factoring out the greatest common factor (GCF) of terms in a polynomial.
2) Factoring trinomials of the form x^2 + bx + c by finding two numbers whose product is c and sum is b.
3) Factoring polynomials using grouping when the GCF is not a single term.
4) Recognizing perfect square trinomials and factoring using (a + b)^2 = a^2 + 2ab + b^2.
The document discusses factoring polynomials. It begins by outlining Swartz's steps for factoring: 1) factor out the greatest common factor (GCF), 2) factor based on the number of terms using techniques like difference of squares or grouping. It then explains how to find the GCF of integers or terms. Several examples are provided of factoring polynomials by finding the GCF and using techniques like difference of squares, grouping, or recognizing perfect square trinomials. Factoring trinomials of the form x^2 + bx + c is also demonstrated.
This document provides information about a college algebra course taught by Mr. Bianco in Fall 2013. It covers topics like multiplying polynomials using FOIL, factoring polynomials by finding the greatest common factor and factoring trinomials. Examples are provided to demonstrate how to factor polynomials of different forms like ax^2 + bx + c. Students are reminded to check their factoring work by multiplying the factors.
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 document outlines key concepts and examples for factoring polynomials. It covers factoring trinomials of the forms x^2 + bx + c, ax^2 + bx + c, and x^2 + bx + c by grouping. Examples are provided to demonstrate finding the greatest common factor of terms, factoring trinomials by finding two numbers whose product and sum meet the given criteria, and checking factoring results using FOIL multiplication. Sections include the greatest common factor, factoring trinomials of the forms x^2 + bx + c and ax^2 + bx + c.
This document outlines key concepts and examples for factoring polynomials. It discusses factoring trinomials of the forms x^2 + bx + c, ax^2 + bx + c, and x^2 + bx + c by grouping. Examples are provided to demonstrate finding the greatest common factor of terms, factoring trinomials by finding two numbers whose product and sum meet the given criteria, and checking factoring results using FOIL. Sections cover the greatest common factor, factoring trinomials of different forms, and solving quadratic equations by factoring.
This document provides an overview of Chapter 13 from a developmental mathematics textbook. It covers factoring polynomials, including:
- Finding the greatest common factor of polynomials
- Factoring trinomials of the form x^2 + bx + c and ax^2 + bx + c
- Checking factored results by multiplying the factors
The chapter sections are described and examples are provided for each type of factoring technique.
This document provides instructions on factoring polynomials of various forms:
1) It explains how to factor polynomials by finding the greatest common factor (GCF).
2) It describes how to factor trinomials of the form x2 + bx + c by finding two numbers whose sum is b and product is c.
3) It shows how to factor trinomials of the form ax2 + bx + c by finding factors of a, c whose products and sums satisfy the polynomial.
This document discusses various methods for factoring polynomials, including:
1) Factoring out the greatest common factor (GCF) of terms in a polynomial.
2) Factoring trinomials of the form x^2 + bx + c by finding two numbers whose product is c and sum is b.
3) Factoring polynomials using grouping when the GCF is not a single term.
4) Recognizing perfect square trinomials and factoring using (a + b)^2 = a^2 + 2ab + b^2.
The document discusses factoring polynomials. It begins by outlining Swartz's steps for factoring: 1) factor out the greatest common factor (GCF), 2) factor based on the number of terms using techniques like difference of squares or grouping. It then explains how to find the GCF of integers or terms. Several examples are provided of factoring polynomials by finding the GCF and using techniques like difference of squares, grouping, or recognizing perfect square trinomials. Factoring trinomials of the form x^2 + bx + c is also demonstrated.
This document provides information about a college algebra course taught by Mr. Bianco in Fall 2013. It covers topics like multiplying polynomials using FOIL, factoring polynomials by finding the greatest common factor and factoring trinomials. Examples are provided to demonstrate how to factor polynomials of different forms like ax^2 + bx + c. Students are reminded to check their factoring work by multiplying the factors.
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 document discusses various techniques for factoring polynomials, including:
1. Factoring using the greatest common factor (GCF).
2. Factoring polynomials with 4 or more terms by grouping.
3. Factoring trinomials using factors that add up to the coefficient of the middle term.
4. Using the "box method" to factor trinomials where the coefficient of the x^2 term is not 1.
5. Factoring the difference of two squares using the formula a^2 - b^2 = (a + b)(a - b).
factoring and the other ones polynomials2.pptScience18
The document discusses various methods for factoring polynomials:
1) Factoring out the greatest common factor (GCF) of each term.
2) Using the difference of squares formula a2 - b2 = (a + b)(a - b).
3) Grouping terms to produce common binomial factors.
4) Reversing the FOIL process to factor trinomials of the form x2 + bx + c.
5) Factoring polynomials completely by factoring out the GCF and then factoring the remaining terms.
1. The document outlines the daily lesson plan which includes a warm-up on linear vs quadratic equations, reviewing factoring quadratics and special products like difference of squares.
2. The class will focus on factoring polynomials completely using steps like finding the greatest common factor, looking for special cases like difference of squares, and factoring trinomials.
3. Students are assigned class work from section 3.9 in their notebooks due by Friday which contains multiple factoring problems, including using special products like difference of squares.
1) The sum of 2 binomials is 5x^2 - 6x. One binomial is 3x^2 - 2x. The problem is to find the other binomial.
2) The document discusses factoring polynomials by finding the greatest common factor (GCF), grouping, difference of squares, sum/difference of cubes, and special cases.
3) Examples are provided for finding the GCF of numbers, monomials, and factoring polynomials using various methods.
This document provides an overview of various techniques for factoring polynomials, including:
1) Factoring out the greatest common factor (GCF);
2) Factoring by grouping terms with a common factor;
3) Factoring perfect square trinomials where the first and last terms are perfect squares;
4) Factoring trinomials using techniques like the reverse box method or grouping.
The document discusses factoring polynomials and finding the roots of polynomial equations. It defines polynomials and polynomial equations. It then covers several methods for factoring polynomials, including factoring polynomials with a common monomial factor, factoring polynomials that are a difference of squares, factoring trinomials, and factoring polynomials by grouping. It also discusses using the factors to find the solutions or roots of a polynomial equation, which are also known as the zeros of a polynomial function.
The document discusses factoring polynomials by finding the greatest common factor (GCF). It provides examples of factoring polynomials by finding the GCF of the numerical coefficients and variable terms. Students are then given practice problems to factor polynomials by finding the GCF and writing the factored form.
This document provides an overview of factoring polynomials. It begins by explaining that factoring polynomials is an important skill, as it is often the first step in solving problems in later chapters. It then defines factoring as determining what was multiplied to obtain a given quantity. Several methods of factoring polynomials are described, including: factoring out the greatest common factor; factoring by grouping; and factoring quadratic polynomials into two linear factors. Step-by-step examples are provided to illustrate each method.
This document provides examples and instructions for factoring polynomials completely using different factoring methods such as greatest common factor, difference of squares, and grouping. It demonstrates how to determine if a polynomial is already fully factored or if additional factoring is possible. Students are shown step-by-step how to choose the appropriate factoring technique and combine methods as needed to fully factor polynomials that include multiple terms and variables. Practice problems with solutions are included to help students apply the factoring skills.
The document provides an agenda and lesson plan for a math class on Tuesday, November 10th. The agenda includes doing homework problems, a daily scribe activity, and lessons on prime factorization, determining if a number is prime or composite, and finding the greatest common factor using prime factorization. Example problems are provided to demonstrate these math concepts.
Today's class will include a warm-up, factoring polynomials using the greatest common factor (GCF) method and factoring by grouping, and Khan Academy assignments due tonight. Students should show all their work to receive credit for class work involving factoring polynomials using GCF and grouping methods.
1. The document discusses factoring polynomials. It covers factoring trinomials of the form x^2 + bx + c by finding two binomial factors with a sum of b and product of c.
2. The steps for factoring completely are: look for the greatest common factor, look for special cases like difference of squares or perfect square trinomial, find two different binomial factors if not in a special form, and factor by grouping if there are 4 terms.
3. Examples show factoring trinomials and using the FOIL method in reverse to factor. Factoring requires understanding the relationship between factors and terms in a polynomial.
This document discusses factoring trinomials of the form x^2 + bx + c. It explains that to factor such trinomials, you need to find two numbers whose sum is b and whose product is c. It then provides examples of factoring different types of trinomials, such as those where the two numbers have the same sign, different signs, or are not factorable. It emphasizes checking your factored results by multiplying them back out to verify they equal the original trinomial. Finally, it provides additional trinomials for the reader to practice factoring.
This document provides instructions for factoring polynomials by finding the greatest common factor (GCF). It explains that to factor a polynomial, you first find the GCF of all the terms, then divide each term by the GCF to reveal the common monomial factor. Several examples are provided of factoring polynomials by finding and dividing out the GCF. Students are then asked to practice factoring additional polynomials using this method.
1) The document discusses different methods for factoring polynomials, including factoring by GCF and factoring by grouping.
2) Factoring by GCF involves finding the greatest common factor of all terms in the polynomial and writing the polynomial as a product with the GCF factored out.
3) Factoring by grouping involves arranging the terms of a four-term polynomial so the first two terms and last two terms each have a common factor, then factoring these common factors out of the pairs of terms.
This document provides examples and instructions for factoring trinomials of the form ax2 + bx + c. It explains that the factors of c must have a sum of b and a product of ac. Several examples are worked through step-by-step to show how to set up a "MAMA table" to find the appropriate factors and then group the terms of the trinomial accordingly. The document also discusses how the signs of the factors change depending on the signs of b and c. Rules for determining the signs in different cases are presented.
This document provides examples and instructions for factoring trinomials of the form ax2 + bx + c. It explains that the factors of c must have a sum of b and a product of ac. Several trinomials are factored step-by-step using a "MAMA" table to find the appropriate factors of c based on this rule. Both positive and negative examples are provided. Guidelines are given for determining the signs of factors based on the signs of b and c. Factoring trinomials is demonstrated to involve grouping terms with the same variables and factoring out the greatest common factor.
The document discusses different methods for factorizing algebraic expressions, including:
1) Taking out a common factor by finding common factors between terms.
2) Using the difference of squares formula a2 - b2 = (a + b)(a - b).
3) Factorizing trinomials of the form x2 + ax + b by finding two numbers whose sum is a and product is b.
4) Factorizing trinomials that are perfect squares by expressing them as the square of a binomial expression.
5) Solving a quadratic equation ax2 + bx + c = 0 by using the quadratic formula.
Grade 8 Mathematics Common Monomial FactoringChristopherRama
The document provides instructions for an individual activity where students must follow 16 directions within 2 minutes to complete tasks like writing their name, drawing shapes, and tapping their desk. It asks what implications the activity might have if directions are not followed properly. The second part of the document provides examples of factoring polynomials using greatest common factor and common monomial factoring methods. It includes practice problems for students to determine the greatest common factor, quotient, and factored form. The document emphasizes following directions and learning different factoring techniques.
The document outlines a factoring unit test schedule for prime factorization, GCF, factoring trinomials of the form x^2 + bx + c, and special products over 4 class periods. It provides examples of factoring trinomials using a table to find two numbers whose product is the constant term and whose sum is the coefficient of the middle term. Students are shown how to determine the signs of the factors based on the signs of the terms and how to check their factoring using FOIL.
Software Engineering and Project Management - Introduction, Modeling Concepts...Prakhyath Rai
Introduction, Modeling Concepts and Class Modeling: What is Object orientation? What is OO development? OO Themes; Evidence for usefulness of OO development; OO modeling history. Modeling
as Design technique: Modeling, abstraction, The Three models. Class Modeling: Object and Class Concept, Link and associations concepts, Generalization and Inheritance, A sample class model, Navigation of class models, and UML diagrams
Building the Analysis Models: Requirement Analysis, Analysis Model Approaches, Data modeling Concepts, Object Oriented Analysis, Scenario-Based Modeling, Flow-Oriented Modeling, class Based Modeling, Creating a Behavioral Model.
This document discusses various techniques for factoring polynomials, including:
1. Factoring using the greatest common factor (GCF).
2. Factoring polynomials with 4 or more terms by grouping.
3. Factoring trinomials using factors that add up to the coefficient of the middle term.
4. Using the "box method" to factor trinomials where the coefficient of the x^2 term is not 1.
5. Factoring the difference of two squares using the formula a^2 - b^2 = (a + b)(a - b).
factoring and the other ones polynomials2.pptScience18
The document discusses various methods for factoring polynomials:
1) Factoring out the greatest common factor (GCF) of each term.
2) Using the difference of squares formula a2 - b2 = (a + b)(a - b).
3) Grouping terms to produce common binomial factors.
4) Reversing the FOIL process to factor trinomials of the form x2 + bx + c.
5) Factoring polynomials completely by factoring out the GCF and then factoring the remaining terms.
1. The document outlines the daily lesson plan which includes a warm-up on linear vs quadratic equations, reviewing factoring quadratics and special products like difference of squares.
2. The class will focus on factoring polynomials completely using steps like finding the greatest common factor, looking for special cases like difference of squares, and factoring trinomials.
3. Students are assigned class work from section 3.9 in their notebooks due by Friday which contains multiple factoring problems, including using special products like difference of squares.
1) The sum of 2 binomials is 5x^2 - 6x. One binomial is 3x^2 - 2x. The problem is to find the other binomial.
2) The document discusses factoring polynomials by finding the greatest common factor (GCF), grouping, difference of squares, sum/difference of cubes, and special cases.
3) Examples are provided for finding the GCF of numbers, monomials, and factoring polynomials using various methods.
This document provides an overview of various techniques for factoring polynomials, including:
1) Factoring out the greatest common factor (GCF);
2) Factoring by grouping terms with a common factor;
3) Factoring perfect square trinomials where the first and last terms are perfect squares;
4) Factoring trinomials using techniques like the reverse box method or grouping.
The document discusses factoring polynomials and finding the roots of polynomial equations. It defines polynomials and polynomial equations. It then covers several methods for factoring polynomials, including factoring polynomials with a common monomial factor, factoring polynomials that are a difference of squares, factoring trinomials, and factoring polynomials by grouping. It also discusses using the factors to find the solutions or roots of a polynomial equation, which are also known as the zeros of a polynomial function.
The document discusses factoring polynomials by finding the greatest common factor (GCF). It provides examples of factoring polynomials by finding the GCF of the numerical coefficients and variable terms. Students are then given practice problems to factor polynomials by finding the GCF and writing the factored form.
This document provides an overview of factoring polynomials. It begins by explaining that factoring polynomials is an important skill, as it is often the first step in solving problems in later chapters. It then defines factoring as determining what was multiplied to obtain a given quantity. Several methods of factoring polynomials are described, including: factoring out the greatest common factor; factoring by grouping; and factoring quadratic polynomials into two linear factors. Step-by-step examples are provided to illustrate each method.
This document provides examples and instructions for factoring polynomials completely using different factoring methods such as greatest common factor, difference of squares, and grouping. It demonstrates how to determine if a polynomial is already fully factored or if additional factoring is possible. Students are shown step-by-step how to choose the appropriate factoring technique and combine methods as needed to fully factor polynomials that include multiple terms and variables. Practice problems with solutions are included to help students apply the factoring skills.
The document provides an agenda and lesson plan for a math class on Tuesday, November 10th. The agenda includes doing homework problems, a daily scribe activity, and lessons on prime factorization, determining if a number is prime or composite, and finding the greatest common factor using prime factorization. Example problems are provided to demonstrate these math concepts.
Today's class will include a warm-up, factoring polynomials using the greatest common factor (GCF) method and factoring by grouping, and Khan Academy assignments due tonight. Students should show all their work to receive credit for class work involving factoring polynomials using GCF and grouping methods.
1. The document discusses factoring polynomials. It covers factoring trinomials of the form x^2 + bx + c by finding two binomial factors with a sum of b and product of c.
2. The steps for factoring completely are: look for the greatest common factor, look for special cases like difference of squares or perfect square trinomial, find two different binomial factors if not in a special form, and factor by grouping if there are 4 terms.
3. Examples show factoring trinomials and using the FOIL method in reverse to factor. Factoring requires understanding the relationship between factors and terms in a polynomial.
This document discusses factoring trinomials of the form x^2 + bx + c. It explains that to factor such trinomials, you need to find two numbers whose sum is b and whose product is c. It then provides examples of factoring different types of trinomials, such as those where the two numbers have the same sign, different signs, or are not factorable. It emphasizes checking your factored results by multiplying them back out to verify they equal the original trinomial. Finally, it provides additional trinomials for the reader to practice factoring.
This document provides instructions for factoring polynomials by finding the greatest common factor (GCF). It explains that to factor a polynomial, you first find the GCF of all the terms, then divide each term by the GCF to reveal the common monomial factor. Several examples are provided of factoring polynomials by finding and dividing out the GCF. Students are then asked to practice factoring additional polynomials using this method.
1) The document discusses different methods for factoring polynomials, including factoring by GCF and factoring by grouping.
2) Factoring by GCF involves finding the greatest common factor of all terms in the polynomial and writing the polynomial as a product with the GCF factored out.
3) Factoring by grouping involves arranging the terms of a four-term polynomial so the first two terms and last two terms each have a common factor, then factoring these common factors out of the pairs of terms.
This document provides examples and instructions for factoring trinomials of the form ax2 + bx + c. It explains that the factors of c must have a sum of b and a product of ac. Several examples are worked through step-by-step to show how to set up a "MAMA table" to find the appropriate factors and then group the terms of the trinomial accordingly. The document also discusses how the signs of the factors change depending on the signs of b and c. Rules for determining the signs in different cases are presented.
This document provides examples and instructions for factoring trinomials of the form ax2 + bx + c. It explains that the factors of c must have a sum of b and a product of ac. Several trinomials are factored step-by-step using a "MAMA" table to find the appropriate factors of c based on this rule. Both positive and negative examples are provided. Guidelines are given for determining the signs of factors based on the signs of b and c. Factoring trinomials is demonstrated to involve grouping terms with the same variables and factoring out the greatest common factor.
The document discusses different methods for factorizing algebraic expressions, including:
1) Taking out a common factor by finding common factors between terms.
2) Using the difference of squares formula a2 - b2 = (a + b)(a - b).
3) Factorizing trinomials of the form x2 + ax + b by finding two numbers whose sum is a and product is b.
4) Factorizing trinomials that are perfect squares by expressing them as the square of a binomial expression.
5) Solving a quadratic equation ax2 + bx + c = 0 by using the quadratic formula.
Grade 8 Mathematics Common Monomial FactoringChristopherRama
The document provides instructions for an individual activity where students must follow 16 directions within 2 minutes to complete tasks like writing their name, drawing shapes, and tapping their desk. It asks what implications the activity might have if directions are not followed properly. The second part of the document provides examples of factoring polynomials using greatest common factor and common monomial factoring methods. It includes practice problems for students to determine the greatest common factor, quotient, and factored form. The document emphasizes following directions and learning different factoring techniques.
The document outlines a factoring unit test schedule for prime factorization, GCF, factoring trinomials of the form x^2 + bx + c, and special products over 4 class periods. It provides examples of factoring trinomials using a table to find two numbers whose product is the constant term and whose sum is the coefficient of the middle term. Students are shown how to determine the signs of the factors based on the signs of the terms and how to check their factoring using FOIL.
Software Engineering and Project Management - Introduction, Modeling Concepts...Prakhyath Rai
Introduction, Modeling Concepts and Class Modeling: What is Object orientation? What is OO development? OO Themes; Evidence for usefulness of OO development; OO modeling history. Modeling
as Design technique: Modeling, abstraction, The Three models. Class Modeling: Object and Class Concept, Link and associations concepts, Generalization and Inheritance, A sample class model, Navigation of class models, and UML diagrams
Building the Analysis Models: Requirement Analysis, Analysis Model Approaches, Data modeling Concepts, Object Oriented Analysis, Scenario-Based Modeling, Flow-Oriented Modeling, class Based Modeling, Creating a Behavioral Model.
Advanced control scheme of doubly fed induction generator for wind turbine us...IJECEIAES
This paper describes a speed control device for generating electrical energy on an electricity network based on the doubly fed induction generator (DFIG) used for wind power conversion systems. At first, a double-fed induction generator model was constructed. A control law is formulated to govern the flow of energy between the stator of a DFIG and the energy network using three types of controllers: proportional integral (PI), sliding mode controller (SMC) and second order sliding mode controller (SOSMC). Their different results in terms of power reference tracking, reaction to unexpected speed fluctuations, sensitivity to perturbations, and resilience against machine parameter alterations are compared. MATLAB/Simulink was used to conduct the simulations for the preceding study. Multiple simulations have shown very satisfying results, and the investigations demonstrate the efficacy and power-enhancing capabilities of the suggested control system.
Electric vehicle and photovoltaic advanced roles in enhancing the financial p...IJECEIAES
Climate change's impact on the planet forced the United Nations and governments to promote green energies and electric transportation. The deployments of photovoltaic (PV) and electric vehicle (EV) systems gained stronger momentum due to their numerous advantages over fossil fuel types. The advantages go beyond sustainability to reach financial support and stability. The work in this paper introduces the hybrid system between PV and EV to support industrial and commercial plants. This paper covers the theoretical framework of the proposed hybrid system including the required equation to complete the cost analysis when PV and EV are present. In addition, the proposed design diagram which sets the priorities and requirements of the system is presented. The proposed approach allows setup to advance their power stability, especially during power outages. The presented information supports researchers and plant owners to complete the necessary analysis while promoting the deployment of clean energy. The result of a case study that represents a dairy milk farmer supports the theoretical works and highlights its advanced benefits to existing plants. The short return on investment of the proposed approach supports the paper's novelty approach for the sustainable electrical system. In addition, the proposed system allows for an isolated power setup without the need for a transmission line which enhances the safety of the electrical network
Digital Twins Computer Networking Paper Presentation.pptxaryanpankaj78
A Digital Twin in computer networking is a virtual representation of a physical network, used to simulate, analyze, and optimize network performance and reliability. It leverages real-time data to enhance network management, predict issues, and improve decision-making processes.
Prediction of Electrical Energy Efficiency Using Information on Consumer's Ac...PriyankaKilaniya
Energy efficiency has been important since the latter part of the last century. The main object of this survey is to determine the energy efficiency knowledge among consumers. Two separate districts in Bangladesh are selected to conduct the survey on households and showrooms about the energy and seller also. The survey uses the data to find some regression equations from which it is easy to predict energy efficiency knowledge. The data is analyzed and calculated based on five important criteria. The initial target was to find some factors that help predict a person's energy efficiency knowledge. From the survey, it is found that the energy efficiency awareness among the people of our country is very low. Relationships between household energy use behaviors are estimated using a unique dataset of about 40 households and 20 showrooms in Bangladesh's Chapainawabganj and Bagerhat districts. Knowledge of energy consumption and energy efficiency technology options is found to be associated with household use of energy conservation practices. Household characteristics also influence household energy use behavior. Younger household cohorts are more likely to adopt energy-efficient technologies and energy conservation practices and place primary importance on energy saving for environmental reasons. Education also influences attitudes toward energy conservation in Bangladesh. Low-education households indicate they primarily save electricity for the environment while high-education households indicate they are motivated by environmental concerns.
Build the Next Generation of Apps with the Einstein 1 Platform.
Rejoignez Philippe Ozil pour une session de workshops qui vous guidera à travers les détails de la plateforme Einstein 1, l'importance des données pour la création d'applications d'intelligence artificielle et les différents outils et technologies que Salesforce propose pour vous apporter tous les bénéfices de l'IA.
VARIABLE FREQUENCY DRIVE. VFDs are widely used in industrial applications for...PIMR BHOPAL
Variable frequency drive .A Variable Frequency Drive (VFD) is an electronic device used to control the speed and torque of an electric motor by varying the frequency and voltage of its power supply. VFDs are widely used in industrial applications for motor control, providing significant energy savings and precise motor operation.
Redefining brain tumor segmentation: a cutting-edge convolutional neural netw...IJECEIAES
Medical image analysis has witnessed significant advancements with deep learning techniques. In the domain of brain tumor segmentation, the ability to
precisely delineate tumor boundaries from magnetic resonance imaging (MRI)
scans holds profound implications for diagnosis. This study presents an ensemble convolutional neural network (CNN) with transfer learning, integrating
the state-of-the-art Deeplabv3+ architecture with the ResNet18 backbone. The
model is rigorously trained and evaluated, exhibiting remarkable performance
metrics, including an impressive global accuracy of 99.286%, a high-class accuracy of 82.191%, a mean intersection over union (IoU) of 79.900%, a weighted
IoU of 98.620%, and a Boundary F1 (BF) score of 83.303%. Notably, a detailed comparative analysis with existing methods showcases the superiority of
our proposed model. These findings underscore the model’s competence in precise brain tumor localization, underscoring its potential to revolutionize medical
image analysis and enhance healthcare outcomes. This research paves the way
for future exploration and optimization of advanced CNN models in medical
imaging, emphasizing addressing false positives and resource efficiency.
Use PyCharm for remote debugging of WSL on a Windo cf5c162d672e4e58b4dde5d797...shadow0702a
This document serves as a comprehensive step-by-step guide on how to effectively use PyCharm for remote debugging of the Windows Subsystem for Linux (WSL) on a local Windows machine. It meticulously outlines several critical steps in the process, starting with the crucial task of enabling permissions, followed by the installation and configuration of WSL.
The guide then proceeds to explain how to set up the SSH service within the WSL environment, an integral part of the process. Alongside this, it also provides detailed instructions on how to modify the inbound rules of the Windows firewall to facilitate the process, ensuring that there are no connectivity issues that could potentially hinder the debugging process.
The document further emphasizes on the importance of checking the connection between the Windows and WSL environments, providing instructions on how to ensure that the connection is optimal and ready for remote debugging.
It also offers an in-depth guide on how to configure the WSL interpreter and files within the PyCharm environment. This is essential for ensuring that the debugging process is set up correctly and that the program can be run effectively within the WSL terminal.
Additionally, the document provides guidance on how to set up breakpoints for debugging, a fundamental aspect of the debugging process which allows the developer to stop the execution of their code at certain points and inspect their program at those stages.
Finally, the document concludes by providing a link to a reference blog. This blog offers additional information and guidance on configuring the remote Python interpreter in PyCharm, providing the reader with a well-rounded understanding of the process.
Rainfall intensity duration frequency curve statistical analysis and modeling...bijceesjournal
Using data from 41 years in Patna’ India’ the study’s goal is to analyze the trends of how often it rains on a weekly, seasonal, and annual basis (1981−2020). First, utilizing the intensity-duration-frequency (IDF) curve and the relationship by statistically analyzing rainfall’ the historical rainfall data set for Patna’ India’ during a 41 year period (1981−2020), was evaluated for its quality. Changes in the hydrologic cycle as a result of increased greenhouse gas emissions are expected to induce variations in the intensity, length, and frequency of precipitation events. One strategy to lessen vulnerability is to quantify probable changes and adapt to them. Techniques such as log-normal, normal, and Gumbel are used (EV-I). Distributions were created with durations of 1, 2, 3, 6, and 24 h and return times of 2, 5, 10, 25, and 100 years. There were also mathematical correlations discovered between rainfall and recurrence interval.
Findings: Based on findings, the Gumbel approach produced the highest intensity values, whereas the other approaches produced values that were close to each other. The data indicates that 461.9 mm of rain fell during the monsoon season’s 301st week. However, it was found that the 29th week had the greatest average rainfall, 92.6 mm. With 952.6 mm on average, the monsoon season saw the highest rainfall. Calculations revealed that the yearly rainfall averaged 1171.1 mm. Using Weibull’s method, the study was subsequently expanded to examine rainfall distribution at different recurrence intervals of 2, 5, 10, and 25 years. Rainfall and recurrence interval mathematical correlations were also developed. Further regression analysis revealed that short wave irrigation, wind direction, wind speed, pressure, relative humidity, and temperature all had a substantial influence on rainfall.
Originality and value: The results of the rainfall IDF curves can provide useful information to policymakers in making appropriate decisions in managing and minimizing floods in the study area.
Software Engineering and Project Management - Software Testing + Agile Method...Prakhyath Rai
Software Testing: A Strategic Approach to Software Testing, Strategic Issues, Test Strategies for Conventional Software, Test Strategies for Object -Oriented Software, Validation Testing, System Testing, The Art of Debugging.
Agile Methodology: Before Agile – Waterfall, Agile Development.
Optimizing Gradle Builds - Gradle DPE Tour Berlin 2024Sinan KOZAK
Sinan from the Delivery Hero mobile infrastructure engineering team shares a deep dive into performance acceleration with Gradle build cache optimizations. Sinan shares their journey into solving complex build-cache problems that affect Gradle builds. By understanding the challenges and solutions found in our journey, we aim to demonstrate the possibilities for faster builds. The case study reveals how overlapping outputs and cache misconfigurations led to significant increases in build times, especially as the project scaled up with numerous modules using Paparazzi tests. The journey from diagnosing to defeating cache issues offers invaluable lessons on maintaining cache integrity without sacrificing functionality.
Generative AI Use cases applications solutions and implementation.pdfmahaffeycheryld
Generative AI solutions encompass a range of capabilities from content creation to complex problem-solving across industries. Implementing generative AI involves identifying specific business needs, developing tailored AI models using techniques like GANs and VAEs, and integrating these models into existing workflows. Data quality and continuous model refinement are crucial for effective implementation. Businesses must also consider ethical implications and ensure transparency in AI decision-making. Generative AI's implementation aims to enhance efficiency, creativity, and innovation by leveraging autonomous generation and sophisticated learning algorithms to meet diverse business challenges.
https://www.leewayhertz.com/generative-ai-use-cases-and-applications/
4. Martin-Gay, Developmental Mathematics 4
13.1 – The Greatest Common Factor
13.2 – Factoring Trinomials of the Form x2 + bx + c
13.3 – Factoring Trinomials of the Form ax2 + bx + c
13.4 – Factoring Trinomials of the Form x2 + bx + c
by Grouping
13.5 – Factoring Perfect Square Trinomials and
Difference of Two Squares
13.6 – Solving Quadratic Equations by Factoring
13.7 – Quadratic Equations and Problem Solving
Chapter Sections
6. Martin-Gay, Developmental Mathematics 6
Factors
Factors (either numbers or polynomials)
When an integer is written as a product of
integers, each of the integers in the product is a
factor of the original number.
When a polynomial is written as a product of
polynomials, each of the polynomials in the
product is a factor of the original polynomial.
Factoring – writing a polynomial as a product of
polynomials.
7. Martin-Gay, Developmental Mathematics 7
Greatest common factor – largest quantity that is a
factor of all the integers or polynomials involved.
Finding the GCF of a List of Integers or Terms
1) Prime factor the numbers.
2) Identify common prime factors.
3) Take the product of all common prime factors.
• If there are no common prime factors, GCF is 1.
Greatest Common Factor
8. Martin-Gay, Developmental Mathematics 8
Find the GCF of each list of numbers.
1) 12 and 8
12 = 2 · 2 · 3
8 = 2 · 2 · 2
So the GCF is 2 · 2 = 4.
2) 7 and 20
7 = 1 · 7
20 = 2 · 2 · 5
There are no common prime factors so the
GCF is 1.
Greatest Common Factor
Example
9. Martin-Gay, Developmental Mathematics 9
Find the GCF of each list of numbers.
1) 6, 8 and 46
6 = 2 · 3
8 = 2 · 2 · 2
46 = 2 · 23
So the GCF is 2.
2) 144, 256 and 300
144 = 2 · 2 · 2 · 3 · 3
256 = 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2
300 = 2 · 2 · 3 · 5 · 5
So the GCF is 2 · 2 = 4.
Greatest Common Factor
Example
10. Martin-Gay, Developmental Mathematics 10
1) x3 and x7
x3 = x · x · x
x7 = x · x · x · x · x · x · x
So the GCF is x · x · x = x3
2) 6x5 and 4x3
6x5 = 2 · 3 · x · x · x
4x3 = 2 · 2 · x · x · x
So the GCF is 2 · x · x · x = 2x3
Find the GCF of each list of terms.
Greatest Common Factor
Example
11. Martin-Gay, Developmental Mathematics 11
Find the GCF of the following list of terms.
a3b2, a2b5 and a4b7
a3b2 = a · a · a · b · b
a2b5 = a · a · b · b · b · b · b
a4b7 = a · a · a · a · b · b · b · b · b · b · b
So the GCF is a · a · b · b = a2b2
Notice that the GCF of terms containing variables will use the
smallest exponent found amongst the individual terms for each
variable.
Greatest Common Factor
Example
12. Martin-Gay, Developmental Mathematics 12
The first step in factoring a polynomial is to
find the GCF of all its terms.
Then we write the polynomial as a product by
factoring out the GCF from all the terms.
The remaining factors in each term will form a
polynomial.
Factoring Polynomials
13. Martin-Gay, Developmental Mathematics 13
Factor out the GCF in each of the following
polynomials.
1) 6x3 – 9x2 + 12x =
3 · x · 2 · x2 – 3 · x · 3 · x + 3 · x · 4 =
3x(2x2 – 3x + 4)
2) 14x3y + 7x2y – 7xy =
7 · x · y · 2 · x2 + 7 · x · y · x – 7 · x · y · 1 =
7xy(2x2 + x – 1)
Factoring out the GCF
Example
14. Martin-Gay, Developmental Mathematics 14
Factor out the GCF in each of the following
polynomials.
1) 6(x + 2) – y(x + 2) =
6 · (x + 2) – y · (x + 2) =
(x + 2)(6 – y)
2) xy(y + 1) – (y + 1) =
xy · (y + 1) – 1 · (y + 1) =
(y + 1)(xy – 1)
Factoring out the GCF
Example
15. Martin-Gay, Developmental Mathematics 15
Remember that factoring out the GCF from the terms of
a polynomial should always be the first step in factoring
a polynomial.
This will usually be followed by additional steps in the
process.
Factor 90 + 15y2 – 18x – 3xy2.
90 + 15y2 – 18x – 3xy2 = 3(30 + 5y2 – 6x – xy2) =
3(5 · 6 + 5 · y2 – 6 · x – x · y2) =
3(5(6 + y2) – x (6 + y2)) =
3(6 + y2)(5 – x)
Factoring
Example
17. Martin-Gay, Developmental Mathematics 17
Factoring Trinomials
Recall by using the FOIL method that
F O I L
(x + 2)(x + 4) = x2 + 4x + 2x + 8
= x2 + 6x + 8
To factor x2 + bx + c into (x + one #)(x + another #),
note that b is the sum of the two numbers and c is the
product of the two numbers.
So we’ll be looking for 2 numbers whose product is
c and whose sum is b.
Note: there are fewer choices for the product, so
that’s why we start there first.
18. Martin-Gay, Developmental Mathematics 18
Factor the polynomial x2 + 13x + 30.
Since our two numbers must have a product of 30 and a
sum of 13, the two numbers must both be positive.
Positive factors of 30 Sum of Factors
1, 30 31
2, 15 17
3, 10 13
Note, there are other factors, but once we find a pair
that works, we do not have to continue searching.
So x2 + 13x + 30 = (x + 3)(x + 10).
Factoring Polynomials
Example
19. Martin-Gay, Developmental Mathematics 19
Factor the polynomial x2 – 11x + 24.
Since our two numbers must have a product of 24 and a
sum of -11, the two numbers must both be negative.
Negative factors of 24 Sum of Factors
– 1, – 24 – 25
– 2, – 12 – 14
– 3, – 8 – 11
So x2 – 11x + 24 = (x – 3)(x – 8).
Factoring Polynomials
Example
20. Martin-Gay, Developmental Mathematics 20
Factor the polynomial x2 – 2x – 35.
Since our two numbers must have a product of – 35 and a
sum of – 2, the two numbers will have to have different signs.
Factors of – 35 Sum of Factors
– 1, 35 34
1, – 35 – 34
– 5, 7 2
5, – 7 – 2
So x2 – 2x – 35 = (x + 5)(x – 7).
Factoring Polynomials
Example
21. Martin-Gay, Developmental Mathematics 21
Factor the polynomial x2 – 6x + 10.
Since our two numbers must have a product of 10 and a
sum of – 6, the two numbers will have to both be negative.
Negative factors of 10 Sum of Factors
– 1, – 10 – 11
– 2, – 5 – 7
Since there is not a factor pair whose sum is – 6,
x2 – 6x +10 is not factorable and we call it a prime
polynomial.
Prime Polynomials
Example
22. Martin-Gay, Developmental Mathematics 22
You should always check your factoring
results by multiplying the factored polynomial
to verify that it is equal to the original
polynomial.
Many times you can detect computational
errors or errors in the signs of your numbers
by checking your results.
Check Your Result!
24. Martin-Gay, Developmental Mathematics 24
Factoring Trinomials
Returning to the FOIL method,
F O I L
(3x + 2)(x + 4) = 3x2 + 12x + 2x + 8
= 3x2 + 14x + 8
To factor ax2 + bx + c into (#1·x + #2)(#3·x + #4), note
that a is the product of the two first coefficients, c is
the product of the two last coefficients and b is the
sum of the products of the outside coefficients and
inside coefficients.
Note that b is the sum of 2 products, not just 2
numbers, as in the last section.
25. Martin-Gay, Developmental Mathematics 25
Factor the polynomial 25x2 + 20x + 4.
Possible factors of 25x2 are {x, 25x} or {5x, 5x}.
Possible factors of 4 are {1, 4} or {2, 2}.
We need to methodically try each pair of factors until we find
a combination that works, or exhaust all of our possible pairs
of factors.
Keep in mind that, because some of our pairs are not identical
factors, we may have to exchange some pairs of factors and
make 2 attempts before we can definitely decide a particular
pair of factors will not work.
Factoring Polynomials
Example
Continued.
26. Martin-Gay, Developmental Mathematics 26
We will be looking for a combination that gives the sum of the
products of the outside terms and the inside terms equal to 20x.
{x, 25x} {1, 4} (x + 1)(25x + 4) 4x 25x 29x
(x + 4)(25x + 1) x 100x 101x
{x, 25x} {2, 2} (x + 2)(25x + 2) 2x 50x 52x
Factors
of 25x2
Resulting
Binomials
Product of
Outside Terms
Product of
Inside Terms
Sum of
Products
Factors
of 4
{5x, 5x} {2, 2} (5x + 2)(5x + 2) 10x 10x 20x
Factoring Polynomials
Example Continued
Continued.
27. Martin-Gay, Developmental Mathematics 27
Check the resulting factorization using the FOIL method.
(5x + 2)(5x + 2) =
= 25x2 + 10x + 10x + 4
5x(5x)
F
+ 5x(2)
O
+ 2(5x)
I
+ 2(2)
L
= 25x2 + 20x + 4
So our final answer when asked to factor 25x2 + 20x + 4
will be (5x + 2)(5x + 2) or (5x + 2)2.
Factoring Polynomials
Example Continued
28. Martin-Gay, Developmental Mathematics 28
Factor the polynomial 21x2 – 41x + 10.
Possible factors of 21x2 are {x, 21x} or {3x, 7x}.
Since the middle term is negative, possible factors of 10
must both be negative: {-1, -10} or {-2, -5}.
We need to methodically try each pair of factors until
we find a combination that works, or exhaust all of our
possible pairs of factors.
Factoring Polynomials
Example
Continued.
29. Martin-Gay, Developmental Mathematics 29
We will be looking for a combination that gives the sum of
the products of the outside terms and the inside terms equal
to 41x.
Factors
of 21x2
Resulting
Binomials
Product of
Outside Terms
Product of
Inside Terms
Sum of
Products
Factors
of 10
{x, 21x}{1, 10}(x – 1)(21x – 10) –10x 21x – 31x
(x – 10)(21x – 1) –x 210x – 211x
{x, 21x} {2, 5} (x – 2)(21x – 5) –5x 42x – 47x
(x – 5)(21x – 2) –2x 105x – 107x
Factoring Polynomials
Example Continued
Continued.
30. Martin-Gay, Developmental Mathematics 30
Factors
of 21x2
Resulting
Binomials
Product of
Outside Terms
Product of
Inside Terms
Sum of
Products
Factors
of 10
(3x – 5)(7x – 2) 6x 35x 41x
{3x, 7x}{1, 10}(3x – 1)(7x – 10) 30x 7x 37x
(3x – 10)(7x – 1) 3x 70x 73x
{3x, 7x} {2, 5} (3x – 2)(7x – 5) 15x 14x 29x
Factoring Polynomials
Example Continued
Continued.
31. Martin-Gay, Developmental Mathematics 31
Check the resulting factorization using the FOIL method.
(3x – 5)(7x – 2) =
= 21x2 – 6x – 35x + 10
3x(7x)
F
+ 3x(-2)
O
- 5(7x)
I
- 5(-2)
L
= 21x2 – 41x + 10
So our final answer when asked to factor 21x2 – 41x + 10
will be (3x – 5)(7x – 2).
Factoring Polynomials
Example Continued
32. Martin-Gay, Developmental Mathematics 32
Factor the polynomial 3x2 – 7x + 6.
The only possible factors for 3 are 1 and 3, so we know that, if
factorable, the polynomial will have to look like (3x )(x )
in factored form, so that the product of the first two terms in the
binomials will be 3x2.
Since the middle term is negative, possible factors of 6 must both
be negative: {1, 6} or { 2, 3}.
We need to methodically try each pair of factors until we find a
combination that works, or exhaust all of our possible pairs of
factors.
Factoring Polynomials
Example
Continued.
33. Martin-Gay, Developmental Mathematics 33
We will be looking for a combination that gives the sum of the
products of the outside terms and the inside terms equal to 7x.
{1, 6} (3x – 1)(x – 6) 18x x 19x
(3x – 6)(x – 1) Common factor so no need to test.
{2, 3} (3x – 2)(x – 3) 9x 2x 11x
(3x – 3)(x – 2) Common factor so no need to test.
Factors
of 6
Resulting
Binomials
Product of
Outside Terms
Product of
Inside Terms
Sum of
Products
Factoring Polynomials
Example Continued
Continued.
34. Martin-Gay, Developmental Mathematics 34
Now we have a problem, because we have
exhausted all possible choices for the factors,
but have not found a pair where the sum of the
products of the outside terms and the inside
terms is –7.
So 3x2 – 7x + 6 is a prime polynomial and will
not factor.
Factoring Polynomials
Example Continued
35. Martin-Gay, Developmental Mathematics 35
Factor the polynomial 6x2y2 – 2xy2 – 60y2.
Remember that the larger the coefficient, the greater the
probability of having multiple pairs of factors to check.
So it is important that you attempt to factor out any
common factors first.
6x2y2 – 2xy2 – 60y2 = 2y2(3x2 – x – 30)
The only possible factors for 3 are 1 and 3, so we know
that, if we can factor the polynomial further, it will have to
look like 2y2(3x )(x ) in factored form.
Factoring Polynomials
Example
Continued.
36. Martin-Gay, Developmental Mathematics 36
Since the product of the last two terms of the binomials
will have to be –30, we know that they must be
different signs.
Possible factors of –30 are {–1, 30}, {1, –30}, {–2, 15},
{2, –15}, {–3, 10}, {3, –10}, {–5, 6} or {5, –6}.
We will be looking for a combination that gives the sum
of the products of the outside terms and the inside terms
equal to –x.
Factoring Polynomials
Example Continued
Continued.
37. Martin-Gay, Developmental Mathematics 37
Factors
of -30
Resulting
Binomials
Product of
Outside Terms
Product of
Inside Terms
Sum of
Products
{-1, 30} (3x – 1)(x + 30) 90x -x 89x
(3x + 30)(x – 1) Common factor so no need to test.
{1, -30} (3x + 1)(x – 30) -90x x -89x
(3x – 30)(x + 1) Common factor so no need to test.
{-2, 15} (3x – 2)(x + 15) 45x -2x 43x
(3x + 15)(x – 2) Common factor so no need to test.
{2, -15} (3x + 2)(x – 15) -45x 2x -43x
(3x – 15)(x + 2) Common factor so no need to test.
Factoring Polynomials
Example Continued
Continued.
38. Martin-Gay, Developmental Mathematics 38
Factors
of –30
Resulting
Binomials
Product of
Outside Terms
Product of
Inside Terms
Sum of
Products
{–3, 10} (3x – 3)(x + 10) Common factor so no need to test.
(3x + 10)(x – 3) –9x 10x x
{3, –10} (3x + 3)(x – 10) Common factor so no need to test.
(3x – 10)(x + 3) 9x –10x –x
Factoring Polynomials
Example Continued
Continued.
39. Martin-Gay, Developmental Mathematics 39
Check the resulting factorization using the FOIL method.
(3x – 10)(x + 3) =
= 3x2 + 9x – 10x – 30
3x(x)
F
+ 3x(3)
O
– 10(x)
I
– 10(3)
L
= 3x2 – x – 30
So our final answer when asked to factor the polynomial
6x2y2 – 2xy2 – 60y2 will be 2y2(3x – 10)(x + 3).
Factoring Polynomials
Example Continued
41. Martin-Gay, Developmental Mathematics 41
Factoring polynomials often involves additional
techniques after initially factoring out the GCF.
One technique is factoring by grouping.
Factor xy + y + 2x + 2 by grouping.
Notice that, although 1 is the GCF for all four
terms of the polynomial, the first 2 terms have a
GCF of y and the last 2 terms have a GCF of 2.
xy + y + 2x + 2 = x · y + 1 · y + 2 · x + 2 · 1 =
y(x + 1) + 2(x + 1) = (x + 1)(y + 2)
Factoring by Grouping
Example
42. Martin-Gay, Developmental Mathematics 42
Factoring a Four-Term Polynomial by Grouping
1) Arrange the terms so that the first two terms have a
common factor and the last two terms have a common
factor.
2) For each pair of terms, use the distributive property to
factor out the pair’s greatest common factor.
3) If there is now a common binomial factor, factor it out.
4) If there is no common binomial factor in step 3, begin
again, rearranging the terms differently.
• If no rearrangement leads to a common binomial
factor, the polynomial cannot be factored.
Factoring by Grouping
44. Martin-Gay, Developmental Mathematics 44
Factor 2x – 9y + 18 – xy by grouping.
Neither pair has a common factor (other than 1).
So, rearrange the order of the factors.
2x + 18 – 9y – xy = 2 · x + 2 · 9 – 9 · y – x · y =
2(x + 9) – y(9 + x) =
2(x + 9) – y(x + 9) = (make sure the factors are identical)
(x + 9)(2 – y)
Factoring by Grouping
Example
46. Martin-Gay, Developmental Mathematics 46
Recall that in our very first example in Section
4.3 we attempted to factor the polynomial
25x2 + 20x + 4.
The result was (5x + 2)2, an example of a
binomial squared.
Any trinomial that factors into a single
binomial squared is called a perfect square
trinomial.
Perfect Square Trinomials
47. Martin-Gay, Developmental Mathematics 47
In the last chapter we learned a shortcut for squaring a
binomial
(a + b)2 = a2 + 2ab + b2
(a – b)2 = a2 – 2ab + b2
So if the first and last terms of our polynomial to be
factored are can be written as expressions squared, and
the middle term of our polynomial is twice the product
of those two expressions, then we can use these two
previous equations to easily factor the polynomial.
a2 + 2ab + b2 = (a + b)2
a2 – 2ab + b2 = (a – b)2
Perfect Square Trinomials
48. Martin-Gay, Developmental Mathematics 48
Factor the polynomial 16x2 – 8xy + y2.
Since the first term, 16x2, can be written as (4x)2, and
the last term, y2 is obviously a square, we check the
middle term.
8xy = 2(4x)(y) (twice the product of the expressions
that are squared to get the first and last terms of the
polynomial)
Therefore 16x2 – 8xy + y2 = (4x – y)2.
Note: You can use FOIL method to verify that the
factorization for the polynomial is accurate.
Perfect Square Trinomials
Example
49. Martin-Gay, Developmental Mathematics 49
Difference of Two Squares
Another shortcut for factoring a trinomial is when we
want to factor the difference of two squares.
a2 – b2 = (a + b)(a – b)
A binomial is the difference of two square if
1.both terms are squares and
2.the signs of the terms are different.
9x2 – 25y2
– c4 + d4
50. Martin-Gay, Developmental Mathematics 50
Difference of Two Squares
Example
Factor the polynomial x2 – 9.
The first term is a square and the last term, 9, can be
written as 32. The signs of each term are different, so
we have the difference of two squares
Therefore x2 – 9 = (x – 3)(x + 3).
Note: You can use FOIL method to verify that the
factorization for the polynomial is accurate.
52. Martin-Gay, Developmental Mathematics 52
Zero Factor Theorem
Quadratic Equations
•Can be written in the form ax2 + bx + c = 0.
•a, b and c are real numbers and a 0.
•This is referred to as standard form.
Zero Factor Theorem
•If a and b are real numbers and ab = 0, then a = 0
or b = 0.
•This theorem is very useful in solving quadratic
equations.
53. Martin-Gay, Developmental Mathematics 53
Steps for Solving a Quadratic Equation by
Factoring
1) Write the equation in standard form.
2) Factor the quadratic completely.
3) Set each factor containing a variable equal to 0.
4) Solve the resulting equations.
5) Check each solution in the original equation.
Solving Quadratic Equations
54. Martin-Gay, Developmental Mathematics 54
Solve x2 – 5x = 24.
• First write the quadratic equation in standard form.
x2 – 5x – 24 = 0
• Now we factor the quadratic using techniques from
the previous sections.
x2 – 5x – 24 = (x – 8)(x + 3) = 0
• We set each factor equal to 0.
x – 8 = 0 or x + 3 = 0, which will simplify to
x = 8 or x = – 3
Solving Quadratic Equations
Example
Continued.
55. Martin-Gay, Developmental Mathematics 55
•Check both possible answers in the original
equation.
82 – 5(8) = 64 – 40 = 24 true
(–3)2 – 5(–3) = 9 – (–15) = 24 true
•So our solutions for x are 8 or –3.
Example Continued
Solving Quadratic Equations
56. Martin-Gay, Developmental Mathematics 56
Solve 4x(8x + 9) = 5
• First write the quadratic equation in standard form.
32x2 + 36x = 5
32x2 + 36x – 5 = 0
• Now we factor the quadratic using techniques from the
previous sections.
32x2 + 36x – 5 = (8x – 1)(4x + 5) = 0
• We set each factor equal to 0.
8x – 1 = 0 or 4x + 5 = 0
Solving Quadratic Equations
Example
Continued.
8x = 1 or 4x = – 5, which simplifies to x = or 5
.
4
1
8
57. Martin-Gay, Developmental Mathematics 57
•Check both possible answers in the original equation.
1 1 1
4 8 9 4 1 9 4 (10) (10) 5
8
1
8
1
8 8 2
true
5 5
4 8 9 4 10 9 4 ( 1) ( 5)( 1) 5
4
5 5
4 4
4
true
• So our solutions for x are or .
8
1
4
5
Example Continued
Solving Quadratic Equations
58. Martin-Gay, Developmental Mathematics 58
Recall that in Chapter 3, we found the x-intercept of
linear equations by letting y = 0 and solving for x.
The same method works for x-intercepts in quadratic
equations.
Note: When the quadratic equation is written in standard
form, the graph is a parabola opening up (when a > 0) or
down (when a < 0), where a is the coefficient of the x2
term.
The intercepts will be where the parabola crosses the
x-axis.
Finding x-intercepts
59. Martin-Gay, Developmental Mathematics 59
Find the x-intercepts of the graph of y = 4x2 + 11x + 6.
The equation is already written in standard form, so
we let y = 0, then factor the quadratic in x.
0 = 4x2 + 11x + 6 = (4x + 3)(x + 2)
We set each factor equal to 0 and solve for x.
4x + 3 = 0 or x + 2 = 0
4x = –3 or x = –2
x = –¾ or x = –2
So the x-intercepts are the points (–¾, 0) and (–2, 0).
Finding x-intercepts
Example
61. Martin-Gay, Developmental Mathematics 61
Strategy for Problem Solving
General Strategy for Problem Solving
1) Understand the problem
• Read and reread the problem
• Choose a variable to represent the unknown
• Construct a drawing, whenever possible
• Propose a solution and check
2) Translate the problem into an equation
3) Solve the equation
4) Interpret the result
• Check proposed solution in problem
• State your conclusion
62. Martin-Gay, Developmental Mathematics 62
The product of two consecutive positive integers is 132. Find the
two integers.
1.) Understand
Read and reread the problem. If we let
x = one of the unknown positive integers, then
x + 1 = the next consecutive positive integer.
Finding an Unknown Number
Example
Continued
63. Martin-Gay, Developmental Mathematics 63
Finding an Unknown Number
Example continued
2.) Translate
Continued
two consecutive positive integers
x (x + 1)
is
=
132
132
•
The product of
64. Martin-Gay, Developmental Mathematics 64
Finding an Unknown Number
Example continued
3.) Solve
Continued
x(x + 1) = 132
x2 + x = 132 (Distributive property)
x2 + x – 132 = 0 (Write quadratic in standard form)
(x + 12)(x – 11) = 0 (Factor quadratic polynomial)
x + 12 = 0 or x – 11 = 0 (Set factors equal to 0)
x = –12 or x = 11 (Solve each factor for x)
65. Martin-Gay, Developmental Mathematics 65
Finding an Unknown Number
Example continued
4.) Interpret
Check: Remember that x is suppose to represent a positive
integer. So, although x = -12 satisfies our equation, it cannot be a
solution for the problem we were presented.
If we let x = 11, then x + 1 = 12. The product of the two numbers
is 11 · 12 = 132, our desired result.
State: The two positive integers are 11 and 12.
66. Martin-Gay, Developmental Mathematics 66
Pythagorean Theorem
In a right triangle, the sum of the squares of the
lengths of the two legs is equal to the square of the
length of the hypotenuse.
(leg a)2 + (leg b)2 = (hypotenuse)2
leg a
hypotenuse
leg b
The Pythagorean Theorem
67. Martin-Gay, Developmental Mathematics 67
Find the length of the shorter leg of a right triangle if the longer leg
is 10 miles more than the shorter leg and the hypotenuse is 10 miles
less than twice the shorter leg.
The Pythagorean Theorem
Example
Continued
1.) Understand
Read and reread the problem. If we let
x = the length of the shorter leg, then
x + 10 = the length of the longer leg and
2x – 10 = the length of the hypotenuse.
x
+ 10
2 - 10
x
x
68. Martin-Gay, Developmental Mathematics 68
The Pythagorean Theorem
Example continued
2.) Translate
Continued
By the Pythagorean Theorem,
(leg a)2 + (leg b)2 = (hypotenuse)2
x2 + (x + 10)2 = (2x – 10)2
3.) Solve
x2 + (x + 10)2 = (2x – 10)2
x2 + x2 + 20x + 100 = 4x2 – 40x + 100 (multiply the binomials)
2x2 + 20x + 100 = 4x2 – 40x + 100 (simplify left side)
x = 0 or x = 30 (set each factor = 0 and solve)
0 = 2x(x – 30) (factor right side)
0 = 2x2 – 60x (subtract 2x2 + 20x + 100 from both sides)
69. Martin-Gay, Developmental Mathematics 69
The Pythagorean Theorem
Example continued
4.) Interpret
Check: Remember that x is suppose to represent the length of
the shorter side. So, although x = 0 satisfies our equation, it
cannot be a solution for the problem we were presented.
If we let x = 30, then x + 10 = 40 and 2x – 10 = 50. Since 302 +
402 = 900 + 1600 = 2500 = 502, the Pythagorean Theorem
checks out.
State: The length of the shorter leg is 30 miles. (Remember that
is all we were asked for in this problem.)