polynomials
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This document defines key terms related to polynomials including monomial, binomial, trinomial, constant, coefficient, and degree. It also describes methods for multiplying polynomials including multiplying monomials, multiplying polynomials using FOIL, squaring binomials, and multiplying the sum and difference of binomials. Finally, it provides examples of dividing monomials.
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1.9 notes
This document contains examples of solving multi-step equations through addition, subtraction, multiplication, and division. It provides 5 warm up problems involving equations with variables, then explains the steps to solve two-step equations by translating words to math, and using operations to isolate the variable. Examples are given showing division and multiplication used to solve equations with more than one operation.
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This document provides instructions for teaching students about factoring quadratic trinomials. It explains that the product of two binomials with a common term, such as (a + b)(a + c), can be expressed by the formula a2 + (b + c)a + bc. This formula results in a quadratic trinomial since it has three terms. The factors of the trinomial are simply the reverse of this formula. Students are guided through examples of factoring various quadratic trinomials and then do a group activity to practice factoring more examples. They then present their work, followed by a quiz and assignment to further their understanding.
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This document provides information about polynomials presented by Lekhamol V R. It begins by stating the objectives of understanding polynomials, which are to determine the degree of a polynomial, classify polynomials, and write polynomials in standard form. It then defines key terms related to polynomials like monomial, polynomial, binomial, trinomial, coefficient, and degree. The document gives examples of finding the degree of monomials and polynomials. It demonstrates how to classify polynomials by degree and number of terms. Finally, it shows how to write polynomials in standard form and add or subtract polynomials by lining up like terms.
Lekhamol VR's Blog
This document provides information about polynomials presented by Lekhamol V R. It defines key terms related to polynomials like monomial, binomial, trinomial, degree, and standard form. It then gives examples of determining the degree of monomials and polynomials, classifying polynomials by degree and number of terms, writing polynomials in standard form, and adding and subtracting polynomials by lining up like terms or stacking the polynomials. The document aims to explain how to work with polynomials through examples.
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The document discusses two methods for canceling out parentheses in equations: FOIL and the distributive property. FOIL stands for "First, Outer, Inner, Last" and involves multiplying the first, outer, inner, and last terms between two sets of parentheses. The distributive property involves distributing numbers outside parentheses to each term inside. Both methods are demonstrated through worked examples of multiplying out expressions like (4x+2)+(3x+1).
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This document provides information about classifying and manipulating polynomials. It defines key polynomial terms like monomial, binomial, trinomial, degree, and standard form. It explains how to determine the degree of a monomial or polynomial, classify polynomials by degree and number of terms, write polynomials in standard form, and add or subtract polynomials by lining up like terms and combining coefficients. Examples are provided to demonstrate these polynomial concepts and procedures.
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1) The document discusses factoring trinomials of the form x^2 + bx + c by using the FOIL method in reverse.
2) It provides examples of factoring trinomials with positive and negative signs using FOIL, and introduces a sign flowchart to determine the signs of the binomial factors.
3) The key steps are to write the trinomial as a product of two binomials, look at the signs to determine the signs in the binomial factors, then guess and check the last two numbers to fully factor the trinomial.
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The document discusses different methods for factoring different types of expressions:
1) For factoring a binomial, the method is to find the greatest common factor (GCF).
2) For the difference of two perfect squares like x^2 - 16, the method is to break it down and write it as (x - 4)(x + 4).
3) For factoring a trinomial where a is not 1, the method is to use the MFDM (multiply, factor, divide, multiply) approach. If a is 1, the method is to use a "diamond" approach.
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This document contains examples of solving multi-step equations through addition, subtraction, multiplication, and division. It provides 5 warm up problems involving equations with variables, then explains the steps to solve two-step equations by translating words to math, and using operations to isolate the variable. Examples are given showing division and multiplication used to solve equations with more than one operation.
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This document provides instructions for teaching students about factoring quadratic trinomials. It explains that the product of two binomials with a common term, such as (a + b)(a + c), can be expressed by the formula a2 + (b + c)a + bc. This formula results in a quadratic trinomial since it has three terms. The factors of the trinomial are simply the reverse of this formula. Students are guided through examples of factoring various quadratic trinomials and then do a group activity to practice factoring more examples. They then present their work, followed by a quiz and assignment to further their understanding.
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This document provides information about polynomials presented by Lekhamol V R. It begins by stating the objectives of understanding polynomials, which are to determine the degree of a polynomial, classify polynomials, and write polynomials in standard form. It then defines key terms related to polynomials like monomial, polynomial, binomial, trinomial, coefficient, and degree. The document gives examples of finding the degree of monomials and polynomials. It demonstrates how to classify polynomials by degree and number of terms. Finally, it shows how to write polynomials in standard form and add or subtract polynomials by lining up like terms.
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This document provides information about polynomials presented by Lekhamol V R. It defines key terms related to polynomials like monomial, binomial, trinomial, degree, and standard form. It then gives examples of determining the degree of monomials and polynomials, classifying polynomials by degree and number of terms, writing polynomials in standard form, and adding and subtracting polynomials by lining up like terms or stacking the polynomials. The document aims to explain how to work with polynomials through examples.
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The document discusses two methods for canceling out parentheses in equations: FOIL and the distributive property. FOIL stands for "First, Outer, Inner, Last" and involves multiplying the first, outer, inner, and last terms between two sets of parentheses. The distributive property involves distributing numbers outside parentheses to each term inside. Both methods are demonstrated through worked examples of multiplying out expressions like (4x+2)+(3x+1).
lekha'sblog
This document provides information about classifying and manipulating polynomials. It defines key polynomial terms like monomial, binomial, trinomial, degree, and standard form. It explains how to determine the degree of a monomial or polynomial, classify polynomials by degree and number of terms, write polynomials in standard form, and add or subtract polynomials by lining up like terms and combining coefficients. Examples are provided to demonstrate these polynomial concepts and procedures.
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1) The document discusses factoring trinomials of the form x^2 + bx + c by using the FOIL method in reverse.
2) It provides examples of factoring trinomials with positive and negative signs using FOIL, and introduces a sign flowchart to determine the signs of the binomial factors.
3) The key steps are to write the trinomial as a product of two binomials, look at the signs to determine the signs in the binomial factors, then guess and check the last two numbers to fully factor the trinomial.
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The document discusses different methods for factoring different types of expressions:
1) For factoring a binomial, the method is to find the greatest common factor (GCF).
2) For the difference of two perfect squares like x^2 - 16, the method is to break it down and write it as (x - 4)(x + 4).
3) For factoring a trinomial where a is not 1, the method is to use the MFDM (multiply, factor, divide, multiply) approach. If a is 1, the method is to use a "diamond" approach.
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This document provides instruction on solving quadratic equations by factoring. It begins by explaining what factoring is and how to factor trinomials of the form x^2 + bx + c. Examples are given of factoring different types of trinomials, including those where b is negative or c is negative. The document then explains how to use the zero product property to solve quadratic equations after factoring them into binomial form. Several examples of solving equations by factoring are shown. Finally, applications of factoring quadratic equations to area word problems are presented.
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This document discusses how to classify polynomials based on the number of terms and degree. Polynomials are named according to whether they have 1, 2, or 3 terms, making them monomials, binomials, or trinomials respectively. The degree of a polynomial is the greatest exponent of any term. Various examples of polynomials are given and classified based on these naming conventions.
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1. Multiply a polynomial by a monomial by distributing the monomial to each term of the polynomial.
2. Multiply binomials using FOIL - multiplying the first, outer, inner, and last terms.
3. Multiply polynomials by distributing and combining like terms.
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The document provides examples of using properties of numbers and algebra to simplify expressions. It gives step-by-step workings for expressions like 11(10-8), 6(4x+5), and 3⋅5⋅3⋅4. It defines the commutative and associative properties. Later examples show how to write an algebraic expression from a word problem and then simplify it using the distributive property.
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The document discusses combinations and the binomial theorem. It provides formulas for combinations and explains how to use Pascal's triangle to determine the coefficients in binomial expansions. Examples are worked out expanding (2x + 1)^4, (x - 2y)^3, and finding the coefficient of x^4 in (2x - 7)^9. The key points are that combinations involve choosing objects without regard to order, while permutations consider order; Pascal's triangle determines the coefficients in binomial expansions; and the powers of the terms follow a pattern of the first term decreasing and the second increasing.
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The document provides solutions to 7 cryptarithm problems. For each problem, it states the values that letters A and B can represent, then uses logical deductions based on place values to determine the values of A and B that satisfy the cryptarithm equation. The last problem shows that cryptarithm 4 x AB = CAB has no solution because there is no digit whose units place is equal to itself multiplied by 4.
Expresiones algebraicas
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The document provides information on finding the highest common factor (HCF) and lowest common multiple (LCM) of numbers. It defines HCF as the greatest number that divides two or more numbers, and LCM as the smallest number divisible by all numbers. Methods for finding HCF and LCM include factorization and division. Several examples are given to demonstrate calculating HCF and LCM of integers, fractions, and decimals.
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This document discusses solving absolute value equations. It provides 3 steps for solving equations containing absolute value: 1) isolate the absolute value bars, 2) make two equations setting the absolute value equal to the positive and negative answers, and 3) solve each equation. Examples are worked through demonstrating that absolute value equations can have more than one solution. The document also notes that if solving the equations results in the absolute value being equal to a negative number, then there is no solution or an empty set.
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This document discusses different types of factoring expressions including trinomials, binomials, and polynomials. Trinomials have three terms and require factoring out the greatest common factor. Binomials can be factored in four ways and using the FOIL method when multiplying two binomials. Polynomials are expressions with constants, variables, and exponents combined using addition, subtraction, and multiplication.
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multiplying polynomials 2 .pptx
The document discusses four objectives for multiplying polynomials:
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2. Operations on polynomials are covered, including removing grouping symbols, addition, subtraction, and multiplying polynomials.
3. Polynomials can be classified based on degree, number of terms, and type of numerical coefficients. They can be in the standard form of highest degree term to constant term.
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- Multiplying monomials, polynomials, and binomial expressions using rules like distributing terms
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- Evaluating numerical values of expressions by substituting values for variables
- Special multiplication rules for perfect square binomials, difference of squares, and conjugates
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March 18, 2015
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.
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This document provides an outline for teaching various factoring techniques. It begins with definitions of algebraic expressions, polynomials, factors, and factoring. It then covers finding the greatest common factor, factoring by using the GCF, factoring by grouping, factoring differences of squares, factoring perfect square trinomials using the special formula, and solving word problems using factoring. Examples are provided for each technique to demonstrate how to factor different polynomial expressions. Special cases like the sum and difference of cubes are also discussed. The document concludes with an explanation of the quadratic formula.
Factoring polynomials
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.
Perfect square of Binomials
1. The square of a binomial (a + b) is a trinomial with terms a2, 2ab, and b2.
2. To square a binomial, square each term and multiply the unlike terms by 2.
3. Examples are provided of squaring binomials like (x + 6)2 = x2 + 12x + 36 and factoring trinomials into perfect square forms like (x - 2)2.
Foil
The document provides instructions on using the FOIL method to multiply binomials. It explains that FOIL is an acronym that stands for First, Outer, Inner, Last - representing the terms to multiply when multiplying two binomials. Several examples are worked out step-by-step using the FOIL method, showing the multiplication of the first, outer, inner, and last terms and combining like terms.
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3-Special Factoring.ppt
This document summarizes techniques for factoring polynomials that are differences of squares, perfect square trinomials, differences of cubes, and sums of cubes. It includes examples and step-by-step solutions for factoring polynomials of each type. The objectives are to learn how to factor polynomials that are differences of squares, perfect square trinomials, differences of cubes, and sums of cubes. Examples are provided to illustrate each technique.
9.6
The document provides instructions on how to factor quadratic expressions of the form x^2 + bx + c by making a T-table to find two numbers whose sum is b and product is c. It includes examples of factoring quadratic expressions when b and c are positive, when b is negative and c is positive, and when b is positive and c is negative. It also includes an example of solving a multi-step word problem involving factoring a quadratic expression.
Tema 1 Repaso productos notables
1. The document provides a review of notable algebraic products, including definitions, rules, and examples of:
- The square of the sum of two quantities
- The square of the difference of two quantities
- The product of the sum and difference of two quantities
- The cube of the sum of a binomial
- The cube of the difference of a binomial
- The product of two binomials with a common term
2. Students are provided with practice problems applying these notable product rules and asked to identify which rule each problem falls under.
3. Links are provided for an online quiz and practice test for students to evaluate their understanding of the material.
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This document defines key terms related to polynomials including monomial, binomial, trinomial, constant, coefficient, and degree. It also describes methods for multiplying polynomials including multiplying monomials, multiplying polynomials using FOIL, squaring binomials, and multiplying the sum and difference of binomials. Finally, it provides examples of dividing monomials.
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This document defines key terms related to polynomials including monomial, binomial, trinomial, constant, coefficient, and degree. It also describes methods for multiplying polynomials including multiplying monomials, multiplying polynomials using FOIL, squaring binomials, and multiplying the sum and difference of binomials. Finally, it provides examples of dividing monomials.
POLYNOMIALS
This document defines key terms related to polynomials including monomial, binomial, trinomial, constant, coefficient, and degree. It also describes methods for multiplying polynomials including multiplying monomials, multiplying polynomials using FOIL, squaring binomials, and multiplying the sum and difference of binomials. Finally, it provides examples of dividing monomials.
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polynomials
- 1. Submitted by: Litty Cherian class:1BEd Option: Mathematics Roll No:22 Mangalam College of Education Submitted to: Mrs.Sijji Jose Dept. Educational technology
- 2. DEFINITIONS Term A number or a product of a number and variables raised to a power. eg : 3, 5𝑥2, -2𝑥, 9𝑥2y. Coefficient The numerical factor of each term. eg :5𝑥2 , -2𝑥 ,9𝑥2 𝑦. Constant The term without a variable. eg : 3, -6, 5, 32. Polynomial A finite sum of terms of the form axn , where a is a real number and n is a whole number. eg : -15x2 + 2x2 .
- 3. Monomial A polynomial with exactly one term. eg: 𝑎𝑥2 , 2𝑥4 , -9m, 9𝑥2 y. Binomial A polynomial with exactly two term. eg: 𝑥-8, -2𝑥 + 9𝑥2 y. Trinomial A polynomial with exactly three term. eg: 𝑥2 + 𝑥 − 8, 5𝑥2 + 2𝑥 − 7.
- 4. The Degree of a Term with one variable is the exponent on the variable. The Degree of a Term with more than one variable is the sum of the exponents on the variables. The Degree of a Polynomial is the greatest degree of the terms of the polynomial variables. 2 5x 2, 9m 1 2 7x y 3, 5 4 9mn z 10 3 2 3 7x x 3, 4 2 2 3 2 5 6x y x y x 6
- 6. Multiplying Monomials by Monomials Examples: 10 9x x 2 90x 3 7 8 11x x 10 88x Multiplying Monomials by Polynomials 2 4 4 3x x x 3 4x 2 16x 12x 3 2 5 3 2x x x 5 15x 4 5x 3 10x Examples:
- 7. Multiplying Two Polynomials Examples: 1. 2. 2 5 10 3x x x 3 x 2 10x 3x 2 5x 50x 15 3 x= 2 15x 47x 2 4 5 3 4x x x 3 12x 2 16x 2 3x 4x 15x 20 = 3 12x 2 13x 11x 20
- 9. Multiplying Two Binomials using FOIL First terms Outer terms Inner terms Last terms 3 4x x 2 x 4x 3x 12 2 x 7x 12 7 4x x 4x 28 3x = 2 x 7x = 2 x 28
- 10. Squaring Binomials 2 2 2 2 2 2 2 2 a b a ab b a b a ab b 2 4 5x 2 16x 2 20x 25 2 16 40 25x x Example 1.
- 11. Multiplying the Sum and Difference of Two Binomials 2 2 a b a b a b Examples 1. 2. 9 9x x 9x2 x 9x 81 2 81x 88 xx 642 x
- 12. Dividing by a Monomial where 0 a b a b c c c c 8 6 4 3 21 9 12 3 x x x x 8 3 21 3 x x 6 3 9 3 x x 4 3 12 3 x x 5 7x 3 3x 4x Examples: 1.