This document provides an overview of algebraic expressions and operations on them, including:
- Summation, subtraction, multiplication, and division of algebraic expressions
- Obtaining the numeric value of an expression
- Notable products and factoring expressions using notable products
- Examples are provided to demonstrate each concept.
This document discusses various topics related to algebraic expressions and their manipulation, including:
1) Summation of algebraic expressions, covering rules for summing monomials and polynomials.
2) Subtraction of algebraic expressions, following similar rules as summation.
3) Finding the numeric value of algebraic expressions by substituting values for variables.
4) Multiplication and division of algebraic expressions, following rules regarding exponents.
5) Notable products, which allow simplifying algebraic expressions using set rules without full calculation.
Suma, resta, multiplicación y división de expresiones algebraicas. La suma y resta de monomios y polinomios siguen las mismas reglas que las operaciones numéricas. La multiplicación requiere aplicar las propiedades de las potencias y la distribución. Existen productos notables cuya factorización se obtiene por inspección. La factorización transforma expresiones en productos de factores.
The document defines algebraic expressions and discusses various algebraic operations such as addition, subtraction, multiplication, division, and factorization of algebraic expressions. It provides examples to illustrate each operation. Factorization is described as expressing a complicated polynomial as the product of simpler polynomial factors. Common factoring techniques are mentioned, including factoring the difference of squares and factoring trinomials.
Unidad 1 LENGUAJE ALGEBRAICO Y PENSAMIENTO FUNCIONALAndrés Betancur
The document discusses algebraic expressions and operations involving polynomials. It defines variables, monomials, polynomials, and rational expressions. It describes how to classify, add, subtract, multiply, and divide polynomials. It also covers factoring polynomials using the greatest common factor and factoring trinomials. Finally, it discusses how to simplify, add, subtract, multiply and divide rational expressions.
This document defines and explains several concepts in mathematics including real numbers, absolute value, inequalities, sets, and set operations. It discusses how real numbers can be rational or irrational based on whether they have a periodic or non-periodic decimal expansion. Absolute value is defined as the distance from zero on the number line, and properties like non-negativity and the triangular inequality are covered. Inequalities and their properties like reflexivity and symmetry are also outlined. Sets are defined as collections of elements that share properties, and set operations like intersection, union, difference and complement are discussed. Examples are provided throughout to illustrate each concept.
-Suma, Resta y Valor Numérico de Expresiones Algebraicas
-Multiplicación y División de Expresiones Algebraicas
-Productos Notables de Expresiones Algebraicas
-Factorización por Productos Notables
This document defines and explains real numbers and their properties. It discusses:
- The definition and classification of real numbers into natural numbers, integers, rational numbers, and irrational numbers.
- Properties of real numbers under addition, subtraction, multiplication, and division.
- Inequalities and their properties when combining real numbers.
- Absolute value and properties of absolute value inequalities.
This document discusses various topics related to algebraic expressions and their manipulation, including:
1) Summation of algebraic expressions, covering rules for summing monomials and polynomials.
2) Subtraction of algebraic expressions, following similar rules as summation.
3) Finding the numeric value of algebraic expressions by substituting values for variables.
4) Multiplication and division of algebraic expressions, following rules regarding exponents.
5) Notable products, which allow simplifying algebraic expressions using set rules without full calculation.
Suma, resta, multiplicación y división de expresiones algebraicas. La suma y resta de monomios y polinomios siguen las mismas reglas que las operaciones numéricas. La multiplicación requiere aplicar las propiedades de las potencias y la distribución. Existen productos notables cuya factorización se obtiene por inspección. La factorización transforma expresiones en productos de factores.
The document defines algebraic expressions and discusses various algebraic operations such as addition, subtraction, multiplication, division, and factorization of algebraic expressions. It provides examples to illustrate each operation. Factorization is described as expressing a complicated polynomial as the product of simpler polynomial factors. Common factoring techniques are mentioned, including factoring the difference of squares and factoring trinomials.
Unidad 1 LENGUAJE ALGEBRAICO Y PENSAMIENTO FUNCIONALAndrés Betancur
The document discusses algebraic expressions and operations involving polynomials. It defines variables, monomials, polynomials, and rational expressions. It describes how to classify, add, subtract, multiply, and divide polynomials. It also covers factoring polynomials using the greatest common factor and factoring trinomials. Finally, it discusses how to simplify, add, subtract, multiply and divide rational expressions.
This document defines and explains several concepts in mathematics including real numbers, absolute value, inequalities, sets, and set operations. It discusses how real numbers can be rational or irrational based on whether they have a periodic or non-periodic decimal expansion. Absolute value is defined as the distance from zero on the number line, and properties like non-negativity and the triangular inequality are covered. Inequalities and their properties like reflexivity and symmetry are also outlined. Sets are defined as collections of elements that share properties, and set operations like intersection, union, difference and complement are discussed. Examples are provided throughout to illustrate each concept.
-Suma, Resta y Valor Numérico de Expresiones Algebraicas
-Multiplicación y División de Expresiones Algebraicas
-Productos Notables de Expresiones Algebraicas
-Factorización por Productos Notables
This document defines and explains real numbers and their properties. It discusses:
- The definition and classification of real numbers into natural numbers, integers, rational numbers, and irrational numbers.
- Properties of real numbers under addition, subtraction, multiplication, and division.
- Inequalities and their properties when combining real numbers.
- Absolute value and properties of absolute value inequalities.
This document provides an overview of the key concepts covered in Chapter 6 of a mathematics textbook, which includes:
1) Defining rational expressions as ratios of two polynomials and focusing on adding, subtracting, multiplying, and dividing rational expressions.
2) Concluding with solving rational equations and applications of rational equations.
3) Summarizing the chapter sections, which cover topics like rational expressions and functions, operations on rational expressions, and applications of rational equations.
1) The document discusses different types of algebraic expressions such as monomials, binomials, trinomials, and polynomials. It provides examples of each type.
2) It describes how to evaluate algebraic expressions by substituting values for variables and following the proper order of operations.
3) The document presents various operations that can be performed on algebraic expressions such as addition, subtraction, multiplication, division, and factoring. It provides examples of how to perform each operation.
The document provides examples and explanations for solving simple linear equations and inequalities over rational numbers. It includes examples of determining if a number is a solution to an equation by substitution, using inverse operations like addition and subtraction to isolate variables, and applying equations to word problems involving forces. Step-by-step workings are shown for solving equations and checking solutions.
Expresiones algebraicas katiuska mendez maria santeliz 0403katiuskaMendez3
This document discusses algebraic expressions, factorization, and radicalization. It begins by explaining that algebraic expressions are important to study as part of mathematical development. It then defines key concepts related to algebraic expressions, including addition, subtraction, multiplication, and division of expressions. It also defines factorization and notable products. The document concludes by providing examples of working through sums, differences, products, and factorizations of algebraic expressions.
The document discusses algorithms for relational database schema design and decomposition. It covers topics like dependency preservation, lossless join properties, synthesizing relations in BCNF and 3NF, and dealing with null values. The key algorithms presented are for relational synthesis into 3NF while preserving dependencies, and relational decomposition into BCNF with lossless joins. Testing for lossless join properties and the potential "dangling tuple" problem with null values are also addressed.
The document contains notes from Chapter 1 on algebra topics including: variables, expressions, equations, exponents, order of operations, real numbers, patterns and functions, scatter plots, and measures of central tendency. Lesson topics include using variables, exponents and order of operations, exploring real numbers, patterns and functions, scatter plots, and defining mean, median, mode, range and outliers. Examples are provided for writing expressions and equations, simplifying expressions, identifying functions, and calculating mean, median and mode.
This document discusses functions generators and recurrence equations. It provides examples of recurrence equations of first and second order. It also discusses combinatorial numbers like binomial coefficients, Catalan numbers, Stirling numbers, and Bell numbers. It defines these numbers and discusses some of their properties. The document provides examples and references for further reading.
This document provides a curriculum map for 6th grade math for the first quarter. It includes two units of study: Number Sense and Operations, and Fractions, Decimals, Percents, and Ratios. For each unit, it lists the big ideas, essential questions, Arizona state standards, essential knowledge and skills, key vocabulary, and technology resources to be covered. The purpose is to outline the math concepts and skills that will be taught to 6th graders in the first quarter.
7 relational database design algorithms and further dependenciesKumar
This chapter discusses algorithms and dependencies for relational database schema design. It covers:
1. Properties of relational decompositions, including the dependency preservation and lossless join properties. Algorithms are presented for testing these properties.
2. Multivalued dependencies and fourth normal form. The chapter describes how to decompose relations based on multivalued dependencies to achieve 4NF.
3. Join dependencies and fifth normal form. Cyclic dependencies can occur with multi-field primary keys, violating 5NF. The chapter provides an example of decomposing a relation to eliminate a cyclic dependency and achieve 5NF.
This document outlines the 7th grade math curriculum for the Isaac School District over 4 quarters. The first quarter focuses on rational number operations and equations. Key concepts include order of operations, prime factorization, integers, and solving one-step and two-step equations. The second quarter covers algebraic expressions, functions, and data analysis such as graphs, tables, and displays of data. The third quarter emphasizes geometry, measurement, and probability. Key areas are geometric properties, principles, and using probability to predict outcomes.
This document defines and provides examples of working with zero, negative, and fractional exponents. It states that any nonzero expression to the zero power is 1, and any term to a negative power is the reciprocal of that term to a positive power. It then gives examples of simplifying expressions using these definitions and the properties of exponents, such as the product, power, and quotient properties.
The document provides information about radicals, exponents, and equations for an exam. It defines square roots, even roots, cube roots, and odd roots. It explains that the square root of a negative number does not exist in the real number system. Radical expressions are defined in terms of their index, radical sign, and radicand. Rational and irrational radical expressions are also discussed. The document also defines exponential expressions and their bases and exponents. Rules are provided for negative exponents, quotient rule, and rational exponents.
1) The document discusses various algebraic expressions including addition, subtraction, multiplication, and division of algebraic expressions. It also covers evaluating numeric values of expressions and notable products of algebraic expressions.
2) Notable products are special multiplication cases where the result can be obtained through inspection without calculating step-by-step. Examples include difference of squares and perfect square trinomials.
3) Factorization techniques are described including factoring by difference of squares, perfect square trinomials, and second degree trinomials.
Starky perez trabajo de matematica 1 unidadStarkiperez
The document is a report on algebraic expressions from a Venezuelan university program. It defines different types of algebraic expressions like monomials, binomials, and polynomials. It also covers operations on algebraic expressions like addition, subtraction, multiplication, division, factorization, and taking roots. Examples are provided for each topic. The document serves to teach students the key concepts and techniques for working with algebraic expressions.
1. The document discusses algebraic expressions, factorization, and radicalization. It provides examples of algebraic expressions, addition and subtraction of expressions, and finding the numeric value of an expression.
2. Notable products are introduced as special multiplication expressions between algebraic terms whose results can be easily determined without step-by-step calculation.
3. Factorization is described as expressing an algebraic term as the product of other terms called factors, such as factoring the number 20 into the prime numbers 2, 2, and 5.
Suma, Resta y Valor numérico de Expresiones algebraicas.
Multiplicación y División de Expresiones algebraicas.
Productos Notables de Expresiones algebraicas.
Factorización por Productos Notables.
1) The document discusses different types of algebraic expressions such as monomials, binomials, trinomials, and polynomials. It provides examples of each type.
2) It describes how to evaluate algebraic expressions by substituting values for variables and following the proper order of operations.
3) The key operations covered are addition, subtraction, multiplication, division, and factorization of algebraic expressions. Step-by-step examples are provided for each type of operation.
This document discusses various topics related to algebraic expressions and their manipulation, including:
1) Summation of algebraic expressions, covering rules for summing monomials and polynomials.
2) Subtraction of algebraic expressions, following similar rules as summation.
3) Finding the numeric value of algebraic expressions by substituting values for variables.
4) Multiplication and division of algebraic expressions, following rules regarding exponents.
5) Notable products, which allow simplifying algebraic expressions using set rules without full calculation.
Expresiones Algebraicas, Factorizacion y radicacionNeilymarMendoza
This document discusses algebraic expressions, factorization, and radicals. It defines algebraic expressions and describes types of expressions such as monomials, binomials, trinomials, and polynomials. It covers how to add, subtract, multiply, and divide expressions. It also discusses factoring expressions, notable products, and properties of radicals such as extracting roots of products, quotients, and other radicals. It provides examples to illustrate each concept and includes a bibliography of references.
Expresiones Algebraicas, Factorizacion y Radicacion.AngeloAngulo1
This document discusses algebraic expressions, factorization, and radicals. It defines algebraic expressions and describes types of expressions such as monomials, binomials, trinomials, and polynomials. It covers how to add, subtract, multiply, and divide expressions. It also discusses factoring expressions, notable products, and properties of radicals such as extracting roots of products, quotients, and other radicals. It provides examples to illustrate each concept and includes a bibliography of references.
This document provides an overview of the key concepts covered in Chapter 6 of a mathematics textbook, which includes:
1) Defining rational expressions as ratios of two polynomials and focusing on adding, subtracting, multiplying, and dividing rational expressions.
2) Concluding with solving rational equations and applications of rational equations.
3) Summarizing the chapter sections, which cover topics like rational expressions and functions, operations on rational expressions, and applications of rational equations.
1) The document discusses different types of algebraic expressions such as monomials, binomials, trinomials, and polynomials. It provides examples of each type.
2) It describes how to evaluate algebraic expressions by substituting values for variables and following the proper order of operations.
3) The document presents various operations that can be performed on algebraic expressions such as addition, subtraction, multiplication, division, and factoring. It provides examples of how to perform each operation.
The document provides examples and explanations for solving simple linear equations and inequalities over rational numbers. It includes examples of determining if a number is a solution to an equation by substitution, using inverse operations like addition and subtraction to isolate variables, and applying equations to word problems involving forces. Step-by-step workings are shown for solving equations and checking solutions.
Expresiones algebraicas katiuska mendez maria santeliz 0403katiuskaMendez3
This document discusses algebraic expressions, factorization, and radicalization. It begins by explaining that algebraic expressions are important to study as part of mathematical development. It then defines key concepts related to algebraic expressions, including addition, subtraction, multiplication, and division of expressions. It also defines factorization and notable products. The document concludes by providing examples of working through sums, differences, products, and factorizations of algebraic expressions.
The document discusses algorithms for relational database schema design and decomposition. It covers topics like dependency preservation, lossless join properties, synthesizing relations in BCNF and 3NF, and dealing with null values. The key algorithms presented are for relational synthesis into 3NF while preserving dependencies, and relational decomposition into BCNF with lossless joins. Testing for lossless join properties and the potential "dangling tuple" problem with null values are also addressed.
The document contains notes from Chapter 1 on algebra topics including: variables, expressions, equations, exponents, order of operations, real numbers, patterns and functions, scatter plots, and measures of central tendency. Lesson topics include using variables, exponents and order of operations, exploring real numbers, patterns and functions, scatter plots, and defining mean, median, mode, range and outliers. Examples are provided for writing expressions and equations, simplifying expressions, identifying functions, and calculating mean, median and mode.
This document discusses functions generators and recurrence equations. It provides examples of recurrence equations of first and second order. It also discusses combinatorial numbers like binomial coefficients, Catalan numbers, Stirling numbers, and Bell numbers. It defines these numbers and discusses some of their properties. The document provides examples and references for further reading.
This document provides a curriculum map for 6th grade math for the first quarter. It includes two units of study: Number Sense and Operations, and Fractions, Decimals, Percents, and Ratios. For each unit, it lists the big ideas, essential questions, Arizona state standards, essential knowledge and skills, key vocabulary, and technology resources to be covered. The purpose is to outline the math concepts and skills that will be taught to 6th graders in the first quarter.
7 relational database design algorithms and further dependenciesKumar
This chapter discusses algorithms and dependencies for relational database schema design. It covers:
1. Properties of relational decompositions, including the dependency preservation and lossless join properties. Algorithms are presented for testing these properties.
2. Multivalued dependencies and fourth normal form. The chapter describes how to decompose relations based on multivalued dependencies to achieve 4NF.
3. Join dependencies and fifth normal form. Cyclic dependencies can occur with multi-field primary keys, violating 5NF. The chapter provides an example of decomposing a relation to eliminate a cyclic dependency and achieve 5NF.
This document outlines the 7th grade math curriculum for the Isaac School District over 4 quarters. The first quarter focuses on rational number operations and equations. Key concepts include order of operations, prime factorization, integers, and solving one-step and two-step equations. The second quarter covers algebraic expressions, functions, and data analysis such as graphs, tables, and displays of data. The third quarter emphasizes geometry, measurement, and probability. Key areas are geometric properties, principles, and using probability to predict outcomes.
This document defines and provides examples of working with zero, negative, and fractional exponents. It states that any nonzero expression to the zero power is 1, and any term to a negative power is the reciprocal of that term to a positive power. It then gives examples of simplifying expressions using these definitions and the properties of exponents, such as the product, power, and quotient properties.
The document provides information about radicals, exponents, and equations for an exam. It defines square roots, even roots, cube roots, and odd roots. It explains that the square root of a negative number does not exist in the real number system. Radical expressions are defined in terms of their index, radical sign, and radicand. Rational and irrational radical expressions are also discussed. The document also defines exponential expressions and their bases and exponents. Rules are provided for negative exponents, quotient rule, and rational exponents.
1) The document discusses various algebraic expressions including addition, subtraction, multiplication, and division of algebraic expressions. It also covers evaluating numeric values of expressions and notable products of algebraic expressions.
2) Notable products are special multiplication cases where the result can be obtained through inspection without calculating step-by-step. Examples include difference of squares and perfect square trinomials.
3) Factorization techniques are described including factoring by difference of squares, perfect square trinomials, and second degree trinomials.
Starky perez trabajo de matematica 1 unidadStarkiperez
The document is a report on algebraic expressions from a Venezuelan university program. It defines different types of algebraic expressions like monomials, binomials, and polynomials. It also covers operations on algebraic expressions like addition, subtraction, multiplication, division, factorization, and taking roots. Examples are provided for each topic. The document serves to teach students the key concepts and techniques for working with algebraic expressions.
1. The document discusses algebraic expressions, factorization, and radicalization. It provides examples of algebraic expressions, addition and subtraction of expressions, and finding the numeric value of an expression.
2. Notable products are introduced as special multiplication expressions between algebraic terms whose results can be easily determined without step-by-step calculation.
3. Factorization is described as expressing an algebraic term as the product of other terms called factors, such as factoring the number 20 into the prime numbers 2, 2, and 5.
Suma, Resta y Valor numérico de Expresiones algebraicas.
Multiplicación y División de Expresiones algebraicas.
Productos Notables de Expresiones algebraicas.
Factorización por Productos Notables.
1) The document discusses different types of algebraic expressions such as monomials, binomials, trinomials, and polynomials. It provides examples of each type.
2) It describes how to evaluate algebraic expressions by substituting values for variables and following the proper order of operations.
3) The key operations covered are addition, subtraction, multiplication, division, and factorization of algebraic expressions. Step-by-step examples are provided for each type of operation.
This document discusses various topics related to algebraic expressions and their manipulation, including:
1) Summation of algebraic expressions, covering rules for summing monomials and polynomials.
2) Subtraction of algebraic expressions, following similar rules as summation.
3) Finding the numeric value of algebraic expressions by substituting values for variables.
4) Multiplication and division of algebraic expressions, following rules regarding exponents.
5) Notable products, which allow simplifying algebraic expressions using set rules without full calculation.
Expresiones Algebraicas, Factorizacion y radicacionNeilymarMendoza
This document discusses algebraic expressions, factorization, and radicals. It defines algebraic expressions and describes types of expressions such as monomials, binomials, trinomials, and polynomials. It covers how to add, subtract, multiply, and divide expressions. It also discusses factoring expressions, notable products, and properties of radicals such as extracting roots of products, quotients, and other radicals. It provides examples to illustrate each concept and includes a bibliography of references.
Expresiones Algebraicas, Factorizacion y Radicacion.AngeloAngulo1
This document discusses algebraic expressions, factorization, and radicals. It defines algebraic expressions and describes types of expressions such as monomials, binomials, trinomials, and polynomials. It covers how to add, subtract, multiply, and divide expressions. It also discusses factoring expressions, notable products, and properties of radicals such as extracting roots of products, quotients, and other radicals. It provides examples to illustrate each concept and includes a bibliography of references.
An algebraic expression is a combination of letters and numbers linked by operation signs: addition, subtraction, multiplication, division and exponentiation. Algebraic expressions allow us, for example, to find areas and volumes. Some examples given are the circumference of a circle (2πr), the area of a square (s=l2), and the volume of a cube (V=a3). The document then provides examples and explanations of algebraic addition, subtraction, multiplication, division, and factorization.
This document provides information about algebraic expressions in Spanish. It defines monomials, binomials, trinomials, and polynomials. It also discusses evaluating algebraic expressions by substituting values for variables. Additional sections cover adding, subtracting, multiplying, and dividing algebraic expressions. It introduces factoring by using factoring formulas called "productos notables" or notable products.
The document discusses algebraic expressions and their operations. It defines algebraic expressions as combinations of letters, signs, and numbers used in mathematical operations. Letters typically represent unknown quantities called variables. The four fundamental operations covered are addition, subtraction, multiplication, and division of algebraic expressions. It also discusses factoring algebraic expressions using notable products, which allow simplifying expressions through inspection rather than calculating operations.
This document provides an overview of algebraic expressions. It defines algebraic expressions as sets of numbers and letters combined using operations like addition, subtraction, multiplication, division, and parentheses. It describes the main elements of algebraic expressions as variables, coefficients, exponents, operators, and parentheses. It also discusses types of expressions like monomials and polynomials, and algebraic operations like addition, subtraction, multiplication, division, and factoring of expressions using notable products. It includes examples for each topic and a bibliography of additional resources.
This document discusses algebraic expressions, factorization, and radicals in Spanish. It provides examples of adding, subtracting, multiplying, and dividing algebraic expressions. It also covers factoring expressions using common factors and the difference of squares, as well as evaluating expressions numerically by substituting values for variables.
The document is about algebraic operations such as addition, subtraction, multiplication, and division. It provides definitions and examples of each operation. It also discusses factoring expressions and notable products. Specifically, it defines addition as combining two or more algebraic expressions into a single expression. It provides the general rule for adding terms with the same or different signs. It also discusses using numerical values to check the results of algebraic operations.
This document discusses sets and real numbers. It defines sets as collections of objects that have a common characteristic. It describes set operations like union, intersection and difference. It defines real numbers as the collection of rational and irrational numbers. It provides examples of real numbers and discusses problems involving sets and inequalities. The document is intended to teach concepts related to sets, real numbers and the number line.
This document discusses algebraic expressions and methods for working with them. It defines algebraic expressions as combinations of letters and numbers using mathematical operations. Letters represent unknown quantities called variables. It then provides examples of common algebraic expressions and explains how to express word problems in algebraic language. Finally, it covers topics like evaluating algebraic expressions for a given value, adding and multiplying terms, factoring expressions, and finding the numerical value of an expression.
Similar to Matemática. Unidad 1. Gabriela Araque. Sección AD0104 (20)
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 Add Chatter in the odoo 17 ERP ModuleCeline George
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Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
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Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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Main Java[All of the Base Concepts}.docxadhitya5119
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1. UNIDAD I
EXPRESIONES ALGEBRAICAS
G A B R I E L A A R A Q U E O V I E D O
C . I 2 5 . 7 5 3 . 9 11
S E C C I Ó N A D 0 1 0 4
Matemática
PNF Administración
06-02-2021
Gabriela Araque. PNF Administración. UPTAEB
1
2. Contenido
Suma Resta y Valor Numérico de Expresiones Algebraicas
Multiplicación y División de Expresiones Algebraicas
Productos Notables de Expresiones Algebraicas
Factorización por Productos Notables
Bibliografía
06-02-2021
Gabriela Araque. PNF Administración. UPTAEB
2
3. Expresión Algebraica
Expresión Algebraica
Una expresión algebraica es una combinación de letras o letras y números unidos por
medio de las operaciones: suma, resta, multiplicación, división, potenciación o radicación,
de manera finita.
Usualmente las primeras letras de nuestro alfabeto: a, b, c, d, etc. si no se dice otra cosa,
representan valores fijos en la expresión. Estas letras también se pueden
llamar parámetros.
Las últimas letras de nuestro alfabeto: x, y, z, u otros símbolos, representan variables que
pueden tomar valores dentro de un subconjunto de números reales
• Si una expresión algebraica está formada por un solo término se llama monomio.
Ejemplo: 3ax 2
• Si la expresión algebraica tiene varios términos se llama polinomio.
Suma de Expresiones Algebraicas
Para sumar dos o más expresiones algebraicas con uno o más términos, se deben reunir
todos los términos semejantes que existan, en uno sólo. Se puede aplicar la propiedad
distributiva de la multiplicación con respecto de la suma.
06-02-2021
Gabriela Araque. PNF Administración. UPTAEB
3
4. Suma y Resta de Expresiones Algebraicas
Problemas Resueltos
Suma de Binomios
𝟔𝒙𝟐
+ 𝟑𝒙𝟐
Respuesta
𝟔𝒙𝟐 + 𝟑𝒙𝟔 = 𝟗𝒙𝟐
Suma de Polinomios
𝟑𝒙𝟑
− 𝟓𝒙𝟐
+ 𝟑𝐱 + 𝟐
2𝒙𝟑
+ 𝟒𝒙𝟐
− 𝟓𝒙 − 𝟏
Respuesta 𝟓𝒙𝟑
− 𝒙𝟐
− 𝟐𝒙 + 𝟏
Resta de Expresiones Algebraicas
La resta algebraica es una de las operaciones fundamentales en el estudio del álgebra. Sirve para
restar monomios y polinomios. Con la resta algebraica sustraemos el valor de una expresión
algebraica de otra.
06-02-2021
Gabriela Araque. PNF Administración. UPTAEB
4
5. Resta de Expresiones Algebraicas
Valor Numérico
Monomio: Se restan los coeficientes de cada expresión como resultado de sacar como factor común
la parte literal
Polinomio: Se deben reunir y ordenar todos los términos semejantes que existan, y proceder a
efectuar la resta.
Problemas Resueltos
Monomios 𝟐𝒙 − 𝟒𝒙
Respuesta 𝟐𝒙 − 𝟒𝒙 = 𝟐 − 𝟒 𝒙 = −𝟐𝒙
Polinomios 𝟓𝒙𝒚𝟐
+ 𝟔𝒚 + 𝟖𝒘
Restar − 𝟓𝒙𝒚𝟐
+ 𝟑𝒚
Respuesta 𝟎 + 𝟑𝒚 + 𝟖𝒘
Valor Numérico
El valor numérico de una expresión algebraica, es el número que se obtiene al sustituir en ésta por
valor numérico dado y realizar las operaciones indicadas. El valor numérico de un polinomio es el
resultado que obtenemos al sustituir la variable x por un número cualquiera.
Problema Resuelto
P(x) = 2x3 + 5x - 3 ;
Obtener valor numérico para x = 1
Respuesta P(1) = 2 · 13 + 5 · 1 - 3 = 2 + 5 - 3 = 4
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6. Multiplicación de Expresiones Algebraicas
Para multiplicar expresiones algebraicas se deben seguir las propiedades de
las potencias. Para ello, se multiplican los coeficientes, y si se multiplican dos incógnitas,
se suman los exponentes de cada una. Para esta operación se debe de aplicar la regla de los
signos.
Regla de los Signos
Operaciones con Potencias
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7. Multiplicación y División de Expresiones Algebraicas
Problema Resuelto
Multiplicación 𝟓𝒙𝟐
𝟑𝒙 − 𝟕 Multiplicación −𝒙 𝒙𝟐
− 𝟓𝒙 − 𝟑
Respuesta 𝟓𝒙𝟐
𝟑𝒙 − 𝟕 = 𝟏𝟓𝒙𝟑
− 𝟑𝟓𝒙𝟐
Respuesta −𝒙𝟑
+ 𝟓𝒙𝟐
+ 𝟑𝒙
División de Monomios
Para dividir monomios se resta los exponentes de las potencias de misma base siguiendo la ley de los
exponentes. Ahora bien, la potencia de un número es el producto de varios factores iguales a él. El número
que se multiplica por si mismo se llama base de la potencia
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8. División de Expresiones Algebraicas
División de Monomios
División de Monomios
Para dividir monomios se resta los exponentes de las potencias de misma base siguiendo la ley de los
exponentes
𝑎𝑚
𝑎𝑛
Problema
𝟒𝒂𝒙𝟒𝒚𝟑
𝟐𝒙𝟐𝒚
Respuesta
𝟒𝒂𝒙𝟒𝒚𝟑
𝟐𝒙𝟐𝒚
= 𝟐𝒂𝒙𝟐
𝒚𝟐
División de Polinomios entre Monomio
Para dividir un polinomio entre un monomio basta con dividir cada uno de los términos del dividendo
entre el término del divisor. Problema
𝟏𝟐𝒙𝟒𝒚+𝟖𝒙𝟑𝒚−𝟐𝟒𝒙𝟐𝒚
𝟒𝒙𝒚
Respuesta 𝟑𝒙𝟑
+ 𝟐𝒙𝟐
− 𝟔𝒙
División de Polinomio entre Polinomio
Para la división de polinomio entre polinomio se debe considerar ordenar cada término del divisor y el
dividendo con respecto a una letra, considerando el exponente de mayor a menor.
Problema Dividir P(x) = 2x4 +2x -1 entre Q(x) = x2-1
Primero, se ordena el polinomio dividendo, según las potencias decrecientes, y se completa con
monomios nulos, para las potencias faltantes:
P(x) = 2x4 + 0x3 +0x2 +2x -1
Q(x) = x2-1
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9. División de Expresiones Algebraicas
Segundo, se divide el primer termino del dividendo ( 2x4) por el primer término del divisor, (x2) obteniéndose así el
primer término del cociente:𝟐𝒙𝟒
+ 𝟎𝒙𝟑
+ 𝟎𝒙𝟐
+ 𝟐𝒙 − 𝟏 𝒙𝟐
− 𝟏
Tercero, multiplicar este término (2x2 ) por el divisor (x 2 – 1) y cambiarle el signo
Y el producto obtenido ( -2x4 + 2x2 ), se suma al dividendo ( 2x4 + 0x3 + 0x2 + 2x - 1) , y se obtiene de esta
manera, un nuevo dividendo
Cuarto, se reiteran los pasos 2 y 3 tantas veces como sea necesario, hasta que el dividendo se transforme en el
polinomio nulo, o su grado sea menor que el divisor.
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10. División de Expresiones Algebraicas
De esta manera se obtiene
𝟐𝒙𝟒
+ 𝟎𝒙𝟑
+ 𝟎𝒙𝟐
+ 𝟐𝒙 − 𝟏 𝒙𝟐
− 𝟏
−𝟐𝒙𝟒
+ 𝟐𝒙𝟐
𝟐𝒙𝟐
+ 𝟐
𝟐𝒙𝟐
+ 𝟐𝒙 − 𝟏
−𝟐𝒙𝟐
+ 𝟐
𝟐𝒙 + 𝟏
Donde G(x) = 2x2 +2 es el cociente y R(x) = 2x +1 es el resto
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11. Productos Notables de Expresiones Algebraicas
Producto Notable
En matemáticas, un producto corresponde al resultado que se obtiene al realizar una multiplicación.
Sabemos que algo es notable cuando nos llama la atención o destaca entre un grupo de cosas.
Los productos notables son simplemente multiplicaciones especiales entre expresiones algebraicas, que
por sus características destacan de las demás multiplicaciones. Las características que hacen que un producto
sea notable, es que se cumplen ciertas reglas, tal que el resultado puede ser obtenido mediante una simple
inspección, sin la necesidad de verificar o realizar la multiplicación paso a paso.
Los productos notables o identidades notables permiten realizar operaciones con expresiones algebraicas
de una manera mas sencilla; debido a que podemos transformar un polinomio grande en dos polinomios mas
pequeños sin alterar la expresión o polinomio original, usando cualquiera de los tipos de producto notable.
Existe varios tipos de productos notables o identidades notables, cada uno con su característica particular, sus
diferente forma de resolver y con distintas reglas que cumplir, entre estos podemos mencionar los siguientes:
Binomio al cuadrado.
Binomio al cubo.
Binomios conjugados.
Binomios con un termino común.
Trinomio al cuadrado
Trinomio al cubo
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12. Productos Notables de Expresiones Algebraicas
Factorización por Productos Notables
Fórmulas del Binomio al Cuadrado
Formula de suma de un binomio al cuadrado
Formula de resta de un binomio al cuadrado
Formula de suma de un binomio al cubo
Fórmula de resta de un binomio al cubo
Factorización
El objetivo de la factorización es llevar un polinomio complicado y expresarlo como el producto de sus factores
polinomiales simples.
Se llaman factores o divisores de una expresión algebraica a las expresiones algebraicas que multiplicadas entre si
dan como producto la primera expresión.
Problema Factorizar 𝒙𝟐
+ 𝟕𝒙 + 𝟏𝟐
Respuesta 𝒙 + 𝟑 𝒙 + 𝟒
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13. Factorización por Productos Notables
Problema
Procediendo a factorizar la expresión 4y2 + 6x7
Respuesta
Divisores del 4: 1 2 4
Divisores del 6: 1 2 3 6
El factor común es 2
Por lo cual se puede factorizar en
4y2 + 6x7 = 2 (2y2 + 3x7)
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Gabriela Araque. PNF Administración. UPTAEB
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14. Factorización por Productos Notables
Factorizando 3x2 + 2x + 15x + 10
Se agrupa de es forma
(3x2 + 2x ) + (15x + 10 )
Sacar factor común por 1 vez a cada en cada término
x(3x + 2) + 5(3x + 2)
Sacar factor común por 2 vez a cada en cada término y se obtiene
(3x + 2)(x + 5)
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15. Bibliografía
• Pontificia Universidad Javeriana (2021). En línea. Disponible en:
http://proyectos.javerianacali.edu.co/cursos_virtuales/pregrado/matematicas_fundament
ales/Expresiones/Cap2/
Expresiones Algebraicas y Sus Operaciones (2021). En línea. Disponible en:
https://es.slideshare.net/guest5d8d8531/expresiones-algebraicas-y-sus-operaciones-
presentation
Operaciones con Expresiones Algebraicas (2021). En línea. Disponible en:
https://es.plusmaths.com/operaciones-con-expresiones-algebraicas.html
Algebra(2021). En línea. Disponible en:
https://www.profesorenlinea.cl/matematica/AlgebraDivision.htm
Taller de matemática (2021). En línea. Disponible en:
http://prometeo.matem.unam.mx/recursos/Licenciatura/TallerMate_UAM_CUAJIMALPA//s
corm_pla
06-02-2021
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