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.
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.
This document discusses special products of binomials, including:
- (a + b)2 = a2 + 2ab + b2, known as a perfect-square trinomial
- (a - b)2 = a2 - 2ab + b2, also a perfect-square trinomial
- (a + b)(a - b) = a2 - b2, known as the difference of two squares
It provides examples of using these rules to simplify expressions involving binomials squared or multiplied together.
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 will help you in factoring sum and difference of two cubes.
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This document provides examples and rules for working with exponents and polynomials. It begins by showing the step-by-step working of three multiplication problems involving exponents. It then states the product rule for exponents and the rule for raising a power to another power. The document encourages positive thinking and hard work. It ends by having the reader say a phrase aloud together to reinforce a growth mindset towards math.
This document provides instruction on factoring polynomials. It begins with examples of factoring linear and quadratic expressions. It then discusses using the Factor Theorem and the Remainder Theorem to determine if a binomial is a factor of a polynomial. Additional examples demonstrate factoring polynomials by grouping like terms and using special rules to factor the sum and difference of cubes. An example applies these factoring techniques to model the volume of a storage box.
1. There are three techniques for multiplying polynomials: the distributive property, FOIL, and the box method.
2. The distributive property involves distributing terms being multiplied over terms in parentheses. FOIL stands for First, Outer, Inner, Last and involves multiplying the first, outer, inner, and last terms of factors being multiplied.
3. The document provides examples of applying these techniques, such as using FOIL to expand (x+3)(x-3) as x2 - 3x + 3x - 9 = x2 - 9.
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.
This document discusses special products of binomials, including:
- (a + b)2 = a2 + 2ab + b2, known as a perfect-square trinomial
- (a - b)2 = a2 - 2ab + b2, also a perfect-square trinomial
- (a + b)(a - b) = a2 - b2, known as the difference of two squares
It provides examples of using these rules to simplify expressions involving binomials squared or multiplied together.
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 will help you in factoring sum and difference of two cubes.
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This document provides examples and rules for working with exponents and polynomials. It begins by showing the step-by-step working of three multiplication problems involving exponents. It then states the product rule for exponents and the rule for raising a power to another power. The document encourages positive thinking and hard work. It ends by having the reader say a phrase aloud together to reinforce a growth mindset towards math.
This document provides instruction on factoring polynomials. It begins with examples of factoring linear and quadratic expressions. It then discusses using the Factor Theorem and the Remainder Theorem to determine if a binomial is a factor of a polynomial. Additional examples demonstrate factoring polynomials by grouping like terms and using special rules to factor the sum and difference of cubes. An example applies these factoring techniques to model the volume of a storage box.
1. There are three techniques for multiplying polynomials: the distributive property, FOIL, and the box method.
2. The distributive property involves distributing terms being multiplied over terms in parentheses. FOIL stands for First, Outer, Inner, Last and involves multiplying the first, outer, inner, and last terms of factors being multiplied.
3. The document provides examples of applying these techniques, such as using FOIL to expand (x+3)(x-3) as x2 - 3x + 3x - 9 = x2 - 9.
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.
This document discusses working with rational expressions, including:
1) Finding the numbers that must be excluded from the domain to avoid undefined expressions.
2) Simplifying rational expressions by factoring numerators and denominators and cancelling common factors.
3) Multiplying, dividing, adding, and subtracting rational expressions by finding common denominators.
Adding and Subtracting Polynomials - Math 7 Q2W4 LC1Carlo Luna
This document discusses adding and subtracting polynomials. It defines key polynomial terms like monomial, binomial, and trinomial. It explains that when adding or subtracting polynomials, only like terms can be combined by adding or subtracting their coefficients while keeping the variable parts the same. Examples are provided to demonstrate adding and subtracting polynomials, including real-life word problems involving combining polynomial expressions to model total areas or profits. The overall goal is for students to learn how to perform operations on polynomials.
This document provides steps for solving radical equations:
1) Isolate the radical on one side of the equation by performing inverse operations
2) Raise both sides of the equation to a power equal to the index of the radical to eliminate the radical
3) Solve the remaining polynomial equation
It includes examples of solving simpler radical equations as well as more complex equations involving fractions and multiple radicals. Checking solutions is emphasized as extraneous solutions may occasionally occur. Graphing calculators can also help visualize and find solutions to radical equations.
Sum and Difference of Cubes. The sum or difference of two cubes can be factored into a product of a binomial times a trinomial. That is, x3 + y3 = (x + y)(x2 − xy + y2) and x3 − y3 = (x − y)(x2 + xy + y2) .
- The document discusses factoring expressions using the greatest common factor (GCF) method and grouping method.
- With the GCF method, the greatest common factor is determined for each term and factored out.
- The grouping method is used when there are 4 or more terms. The first two terms are factored and the second two terms are factored, then the common factor is extracted.
- Examples of applying both methods to factor polynomials are provided and worked through step-by-step.
rational equation transformable to quadratic equation.pptxRizaCatli2
1. The document provides examples for solving quadratic equations that are not in standard form by transforming them into standard form ax2 + bx + c = 0 and then using methods like factoring or the quadratic formula.
2. It also gives examples for solving rational algebraic equations by multiplying both sides by the least common denominator to obtain a quadratic equation, transforming it into standard form, and then solving.
3. The examples cover topics like solving for the solution set, checking solutions, and using the quadratic formula to solve transformed equations.
This document discusses multiplying polynomials. It begins with examples of multiplying monomials by using the properties of exponents. It then covers multiplying a polynomial by a monomial using the distributive property. Examples are provided for multiplying binomials by binomials using both the distributive property and FOIL method. The document concludes by explaining methods for multiplying polynomials with more than two terms, such as using the distributive property multiple times, a rectangle model, or a vertical method similar to multiplying whole numbers.
Factoring the greatest common monomial factorNara Cocarelli
This document discusses factoring polynomials by finding the greatest common factor (GCF). It explains that factoring a polynomial is the reverse of multiplying a monomial and polynomial. Examples are provided of factoring polynomials by identifying the GCF and rewriting the polynomial as a product of the GCF and other factors. The process of factoring out the GCF is demonstrated step-by-step for several examples.
The document discusses factoring the difference of two squares. It involves reviewing factoring the difference of two squares, which involves recognizing that the difference of two squares can be written as the product of two binomials, where one binomial contains the sum of the two terms and the other contains their difference.
The document defines a polynomial function as a function of the form f(x) = anxn + an-1xn-1 +...+ a0, where n is a nonnegative integer and an, an-1,...a0 are real numbers with an ≠ 0. The degree of a polynomial is the highest exponent of its terms. Examples are provided to illustrate how to determine the degree and number of terms of polynomial functions. The document also asks questions to check understanding of identifying polynomial functions and determining their degree.
1. The document provides examples and explanations for evaluating algebraic expressions by substituting values for variables.
2. It gives examples of evaluating expressions involving addition, subtraction, multiplication, division, and order of operations.
3. Students are asked to evaluate expressions for given variable values to check their understanding.
To multiply polynomials, you can use the distributive property and properties of exponents. When multiplying monomials, group terms with the same bases and add their exponents. When multiplying binomials, use FOIL or distribute one binomial over the other. For polynomials with more than two terms, you can distribute or use a rectangle model to systematically multiply each term.
1. The document discusses multiplying and simplifying monomial expressions. It defines monomials and explains the rules for multiplying them, including adding exponents when multiplying like terms and multiplying all exponents when an expression is raised to a power.
2. The document also discusses multiplying monomials and polynomials using the distributive property and applying the same exponent rules. It provides examples of multiplying and simplifying expressions involving monomials and polynomials.
3. The learning objectives are to multiply monomials, simplify expressions with monomials, and to multiply a monomial and a polynomial.
You will learn how to factor polynomials with common monomial factor.
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This document discusses factoring polynomials. It explains that factoring is reversing the process of multiplication to express a polynomial as a product of binomials. It provides examples of factoring out the greatest common factor (GCF) from a polynomial and factoring trinomials into two binomials. It outlines the step-by-step process of factoring trinomials, emphasizing the importance of checking signs. Finally, it provides examples for the reader to practice factoring polynomials.
This document discusses factoring the sum and difference of two cubes. It explains that the sum or difference of two cubes can be factored into a binomial times a trinomial, with the first term of the trinomial being the cube root of the first term, the second term being the product of the cube roots, and the third term being the cube root of the second term. It provides an example of factoring 27x3 - 125 to show the process.
Factoring Techniques: Difference of Two SquaresCarlo Luna
This document contains notes and exercises on factoring the difference of two squares from a Grade 8 mathematics class. It includes examples of factoring various expressions involving differences of two squares. It also provides guidance on determining when an expression can be factored as a difference of two squares and the steps to follow in factoring them. Several practice exercises are provided for students to try factoring differences of two squares on their own.
solving quadratic equations using quadratic formulamaricel mas
The document discusses how to use the quadratic formula to solve quadratic equations. It provides the formula: x = (-b ± √(b2 - 4ac)) / 2a. It then works through examples of writing quadratic equations in standard form (ax2 + bx + c = 0) and using the formula to solve them. Specifically, it solves the equations: 1) 1x2 + 3x - 27 = 0, 2) 2x2 + 7x + 5 = 0, 3) x2 - 2x = 8, and 4) x2 - 7x = 10. It concludes by providing 3 additional equations to solve using the quadratic formula.
I have added to the original presentation in response to one of the comments.... the result of 'x' is correct on slide 7, take a look at the new version of this ppt to clear up any confusion about why...
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.
1. There are several methods for factoring polynomials outlined in the document: factoring using the distributive property, factoring the difference of two squares/cubes, factoring a perfect square trinomial, and factoring general trinomials using trial and error or grouping.
2. Factoring trinomials involves determining the signs in the factors based on the signs of the terms, then finding two factors of the constant term that satisfy the middle term.
3. More complex polynomials can be factored by grouping like terms or using special factoring patterns like the difference of squares/cubes.
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.
This document discusses working with rational expressions, including:
1) Finding the numbers that must be excluded from the domain to avoid undefined expressions.
2) Simplifying rational expressions by factoring numerators and denominators and cancelling common factors.
3) Multiplying, dividing, adding, and subtracting rational expressions by finding common denominators.
Adding and Subtracting Polynomials - Math 7 Q2W4 LC1Carlo Luna
This document discusses adding and subtracting polynomials. It defines key polynomial terms like monomial, binomial, and trinomial. It explains that when adding or subtracting polynomials, only like terms can be combined by adding or subtracting their coefficients while keeping the variable parts the same. Examples are provided to demonstrate adding and subtracting polynomials, including real-life word problems involving combining polynomial expressions to model total areas or profits. The overall goal is for students to learn how to perform operations on polynomials.
This document provides steps for solving radical equations:
1) Isolate the radical on one side of the equation by performing inverse operations
2) Raise both sides of the equation to a power equal to the index of the radical to eliminate the radical
3) Solve the remaining polynomial equation
It includes examples of solving simpler radical equations as well as more complex equations involving fractions and multiple radicals. Checking solutions is emphasized as extraneous solutions may occasionally occur. Graphing calculators can also help visualize and find solutions to radical equations.
Sum and Difference of Cubes. The sum or difference of two cubes can be factored into a product of a binomial times a trinomial. That is, x3 + y3 = (x + y)(x2 − xy + y2) and x3 − y3 = (x − y)(x2 + xy + y2) .
- The document discusses factoring expressions using the greatest common factor (GCF) method and grouping method.
- With the GCF method, the greatest common factor is determined for each term and factored out.
- The grouping method is used when there are 4 or more terms. The first two terms are factored and the second two terms are factored, then the common factor is extracted.
- Examples of applying both methods to factor polynomials are provided and worked through step-by-step.
rational equation transformable to quadratic equation.pptxRizaCatli2
1. The document provides examples for solving quadratic equations that are not in standard form by transforming them into standard form ax2 + bx + c = 0 and then using methods like factoring or the quadratic formula.
2. It also gives examples for solving rational algebraic equations by multiplying both sides by the least common denominator to obtain a quadratic equation, transforming it into standard form, and then solving.
3. The examples cover topics like solving for the solution set, checking solutions, and using the quadratic formula to solve transformed equations.
This document discusses multiplying polynomials. It begins with examples of multiplying monomials by using the properties of exponents. It then covers multiplying a polynomial by a monomial using the distributive property. Examples are provided for multiplying binomials by binomials using both the distributive property and FOIL method. The document concludes by explaining methods for multiplying polynomials with more than two terms, such as using the distributive property multiple times, a rectangle model, or a vertical method similar to multiplying whole numbers.
Factoring the greatest common monomial factorNara Cocarelli
This document discusses factoring polynomials by finding the greatest common factor (GCF). It explains that factoring a polynomial is the reverse of multiplying a monomial and polynomial. Examples are provided of factoring polynomials by identifying the GCF and rewriting the polynomial as a product of the GCF and other factors. The process of factoring out the GCF is demonstrated step-by-step for several examples.
The document discusses factoring the difference of two squares. It involves reviewing factoring the difference of two squares, which involves recognizing that the difference of two squares can be written as the product of two binomials, where one binomial contains the sum of the two terms and the other contains their difference.
The document defines a polynomial function as a function of the form f(x) = anxn + an-1xn-1 +...+ a0, where n is a nonnegative integer and an, an-1,...a0 are real numbers with an ≠ 0. The degree of a polynomial is the highest exponent of its terms. Examples are provided to illustrate how to determine the degree and number of terms of polynomial functions. The document also asks questions to check understanding of identifying polynomial functions and determining their degree.
1. The document provides examples and explanations for evaluating algebraic expressions by substituting values for variables.
2. It gives examples of evaluating expressions involving addition, subtraction, multiplication, division, and order of operations.
3. Students are asked to evaluate expressions for given variable values to check their understanding.
To multiply polynomials, you can use the distributive property and properties of exponents. When multiplying monomials, group terms with the same bases and add their exponents. When multiplying binomials, use FOIL or distribute one binomial over the other. For polynomials with more than two terms, you can distribute or use a rectangle model to systematically multiply each term.
1. The document discusses multiplying and simplifying monomial expressions. It defines monomials and explains the rules for multiplying them, including adding exponents when multiplying like terms and multiplying all exponents when an expression is raised to a power.
2. The document also discusses multiplying monomials and polynomials using the distributive property and applying the same exponent rules. It provides examples of multiplying and simplifying expressions involving monomials and polynomials.
3. The learning objectives are to multiply monomials, simplify expressions with monomials, and to multiply a monomial and a polynomial.
You will learn how to factor polynomials with common monomial factor.
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This document discusses factoring polynomials. It explains that factoring is reversing the process of multiplication to express a polynomial as a product of binomials. It provides examples of factoring out the greatest common factor (GCF) from a polynomial and factoring trinomials into two binomials. It outlines the step-by-step process of factoring trinomials, emphasizing the importance of checking signs. Finally, it provides examples for the reader to practice factoring polynomials.
This document discusses factoring the sum and difference of two cubes. It explains that the sum or difference of two cubes can be factored into a binomial times a trinomial, with the first term of the trinomial being the cube root of the first term, the second term being the product of the cube roots, and the third term being the cube root of the second term. It provides an example of factoring 27x3 - 125 to show the process.
Factoring Techniques: Difference of Two SquaresCarlo Luna
This document contains notes and exercises on factoring the difference of two squares from a Grade 8 mathematics class. It includes examples of factoring various expressions involving differences of two squares. It also provides guidance on determining when an expression can be factored as a difference of two squares and the steps to follow in factoring them. Several practice exercises are provided for students to try factoring differences of two squares on their own.
solving quadratic equations using quadratic formulamaricel mas
The document discusses how to use the quadratic formula to solve quadratic equations. It provides the formula: x = (-b ± √(b2 - 4ac)) / 2a. It then works through examples of writing quadratic equations in standard form (ax2 + bx + c = 0) and using the formula to solve them. Specifically, it solves the equations: 1) 1x2 + 3x - 27 = 0, 2) 2x2 + 7x + 5 = 0, 3) x2 - 2x = 8, and 4) x2 - 7x = 10. It concludes by providing 3 additional equations to solve using the quadratic formula.
I have added to the original presentation in response to one of the comments.... the result of 'x' is correct on slide 7, take a look at the new version of this ppt to clear up any confusion about why...
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.
1. There are several methods for factoring polynomials outlined in the document: factoring using the distributive property, factoring the difference of two squares/cubes, factoring a perfect square trinomial, and factoring general trinomials using trial and error or grouping.
2. Factoring trinomials involves determining the signs in the factors based on the signs of the terms, then finding two factors of the constant term that satisfy the middle term.
3. More complex polynomials can be factored by grouping like terms or using special factoring patterns like the difference of squares/cubes.
The document defines key polynomial vocabulary including:
- Terms are numbers or products of numbers and variables raised to powers. Coefficients are numerical factors of terms. Constants are terms that are only numbers.
- Polynomials are sums of terms involving variables raised to whole number exponents, with no variables in denominators.
- Types of polynomials include monomials (1 term), binomials (2 terms), and trinomials (3 terms). Degree is the largest exponent of any term.
- Operations on polynomials include adding/subtracting like terms, multiplying using distribution and FOIL, dividing using long division, and special products like (a+b)2 and (a+b)(a
Factoring is the sale of accounts receivable (book debts) by a firm to a financial institution called a factor. The factor provides upfront cash payment for the receivables, usually 80%, and assumes responsibility for collecting payment from customers and managing credit risk. Factoring provides firms with working capital and credit protection. It involves three main parties - the client firm, its customers, and the financial institution factor. Factoring has grown in importance globally as a source of trade financing and working capital for businesses.
This document provides an overview of algebraic expressions and operations with algebraic expressions including:
- Algebraic expressions involve unknown quantities represented by letters combined with numbers and operations.
- The numerical value of an expression is found by substituting values for the unknowns.
- Types of expressions include monomials, binomials, and polynomials.
- Common operations with expressions include addition, multiplication, division, and taking powers of expressions.
- Polynomials can be added, multiplied, and factors can be removed through operations like difference of squares identities.
Like terms are terms whose variables and exponents are the same. To add polynomials, group like terms together and add their coefficients. To multiply polynomials, use the FOIL method: First terms, Outer terms, Inner terms, Last terms.
To summarize the key steps for factoring polynomials:
1. Determine possible integer roots by finding the divisors of the constant term.
2. Use the remainder theorem or Ruffini's rule to check if an integer is a root by dividing the polynomial by (x - a) and checking if the remainder is zero.
3. Integer roots that produce a zero remainder are factors of the polynomial. Repeating this process allows one to fully factor the polynomial into linear terms.
The document discusses polynomials and factoring polynomials. It defines polynomials as expressions with terms added or subtracted, where terms are products of numbers and variables with exponents. It provides examples of monomials, binomials, trinomials, and polynomials based on the number of terms. It also discusses finding the greatest common factor of a polynomial to factor out a monomial.
This document defines polynomials and describes how to perform operations on them such as addition and subtraction. It provides examples of adding and subtracting monomials and polynomials. Monomials are terms with variables and coefficients, and polynomials are the sum of monomials. Like terms refer to monomials with the same variables and exponents that can be combined. To add polynomials, like terms are lined up and their coefficients are summed. To subtract polynomials, the operation is changed to addition by using the keep-change-change method and then like terms are combined.
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.
The document defines the hydrosphere as all the water on Earth, including freshwater, saltwater, groundwater, water vapor, and ice. It notes that the hydrosphere interacts with the atmosphere, lithosphere, and biosphere. Additionally, it states that the hydrosphere has important functions like providing habitat and regulating climate, but is threatened by overuse and pollution.
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.
An exponent or power is a symbol written above and to the right of an expression to indicate raising a base to that power. A product is the answer to a multiplication problem, while a quotient is the answer to a division problem. A coefficient is a number that is multiplied by a variable.
The document discusses measurements, area, right prisms and cylinders, surface area and volume of prisms and cylinders, pyramids, cones and spheres, and how multiplying dimensions by a factor affects surface area and volume. It provides formulas for calculating surface area and volume of various 3D shapes and explains that multiplying the length, breadth, and height by a factor k will result in the surface area being multiplied by k^2 and volume being multiplied by k^3.
Optical phenomena and properties of matterSiyavula
The document discusses optical phenomena and properties of matter in physics. It describes the photoelectric effect where electrons are emitted from a substance when light shines on it. The number of emitted electrons increases with light intensity. It also discusses work function, which is the minimum energy needed to eject electrons from a metal. Emission and absorption spectra are formed when certain light frequencies are emitted or absorbed by gases and materials, respectively, due to electrons moving between energy levels.
This document discusses solving multi-step equations using two or more transformations. It provides 4 cases of multi-step equations as examples: 1) Combining like terms, 2) Using the distributive property, 3) Distributing a negative number, and 4) Multiplying by a reciprocal. For each case, it shows the step-by-step work of solving a sample equation as an example. At the end, it assigns homework problems from the textbook for additional practice.
The document defines and explains different types of functions including linear, quadratic, hyperbolic, exponential, sine, cosine, and tangent functions. It describes how each type of function is represented through equations, tables, graphs and ordered pairs. Key aspects like domains, ranges, periods, asymptotes and effects of variables on the shape of graphs are discussed for each function type.
MNOP is a parallelogram because it has both pairs of opposite sides that are parallel and equal in length. We can use properties of quadrilaterals and geometry to prove shapes are specific types of quadrilaterals. For example, the diagram shows that S is the midpoint of MN, T is the midpoint of NR, so U must be the midpoint of NP. We can prove conjectures about lines and angles using properties we know.
Organic chemistry deals with carbon-containing compounds called organic molecules. Carbon atoms can bond to many other atoms, often forming long chain structures. Organic compounds can be represented using molecular formulas or structural formulas. They contain functional groups that give them certain chemical properties and can be classified based on these groups. Organic reactions include addition, elimination, and substitution. Polymers are large molecules formed by combining repeating structural units (monomers) and include both natural and synthetic varieties.
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 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 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.
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.
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.
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).
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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 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 provides instructions on factorizing quadratic equations. It begins by explaining what quadratic equations are and provides examples. It then discusses factorizing quadratics where the coefficient of x^2 is 1 by finding two numbers whose product is the last term and sum is the middle term. The document continues explaining how to factorize when the coefficient of x^2 is not 1 and predicts the signs of the factors based on the signs of the terms in the quadratic equation. It provides examples of factorizing different quadratic equations.
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.
OpenGL is a software interface for graphics hardware that provides a portable low-level 3D graphics library. It consists of three main libraries - OpenGL (GL) for modeling objects with primitives, OpenGL Utility Library (GLU) for utilities like camera and projection as well as additional modeling functions, and OpenGL Utilities Toolkit (GLUT) for window creation and input handling along with more modeling functions. OpenGL originated from Silicon Graphics' efforts to improve the portability of their IRIS GL graphics API and has evolved through multiple generations to support both fixed and programmable graphics pipelines. It works procedurally by describing graphic rendering steps rather than describing a scene descriptively. Popular programs that use OpenGL include games, 3D modeling software, and virtual
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A researcher tested whether the mean number of days basic automobiles sit on dealers' lots is greater than 29 days. A sample of 30 dealers had a mean of 30.1 days. With a significance level of 0.05 and a population standard deviation of 3.8 days, a one-tailed t-test was conducted. The t-statistic was below the critical value so the null hypothesis that the mean is 29 days was not rejected, meaning there is not enough evidence to say the mean is greater than 29 days.
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Range is defined as the difference between the largest and smallest number or value in a set of data. For the example data set of 4,6,9,3,7, the lowest number is 3, the highest is 9, so the range is calculated as the maximum (9) minus the minimum (3), which equals 6. The document provides a brief definition of range and a simple example calculation but does not include detailed steps for calculating range in Excel.
The document discusses using a t-test to test the difference between means of two independent samples. A t-test can be used when samples are from two normally distributed populations and are independent. The formula for a t-test calculates the test value as the difference between the observed and expected values divided by the standard error. An example compares the average farm sizes in two counties and performs a t-test to determine if the difference in means is statistically significant at the 0.05 level. The t-test results show there is not enough evidence to conclude the average farm sizes are different between the counties.
Long before colonization, the Philippines had its own civilization that was influenced by both Malay settlers and the local environment. Many customs from this pre-colonial society still exist today, providing insights into the country's distant past. Philippine society was organized into barangays led by datus. People lived in extended family groups and practiced traditions related to marriage, religion, and agriculture. While Spanish influence later changed aspects of Philippine culture, the foundations of pre-colonial society remain visible in some modern practices and social structures.
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A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
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Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
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This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
2. Martin-Gay, Developmental Mathematics 2
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
4. Martin-Gay, Developmental Mathematics 4
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.
5. Martin-Gay, Developmental Mathematics 5
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
6. Martin-Gay, Developmental Mathematics 6
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.
1) 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
7. Martin-Gay, Developmental Mathematics 7
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.
1) 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
8. Martin-Gay, Developmental Mathematics 8
1) x3
and x7
x3
= x · x · x
x7
= x · x · x · x · x · x · x
So the GCF is x · x · x = x3
t 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
9. Martin-Gay, Developmental Mathematics 9
Find the GCF of the following list of terms.
a3
b2
, a2
b5
and a4
b7
a3
b2
= a · a · a · b · b
a2
b5
= a · a · b · b · b · b · b
a4
b7
= a · a · a · a · b · b · b · b · b · b · b
So the GCF is a · a · b · b = a2
b2
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
10. Martin-Gay, Developmental Mathematics 10
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
11. Martin-Gay, Developmental Mathematics 11
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) 14x3
y + 7x2
y – 7xy =
7 · x · y · 2 · x2
+ 7 · x · y · x – 7 · x · y · 1 =
7xy(2x2
+ x – 1)
Factoring out the GCF
Example
12. Martin-Gay, Developmental Mathematics 12
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
13. Martin-Gay, Developmental Mathematics 13
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
15. Martin-Gay, Developmental Mathematics 15
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.
16. Martin-Gay, Developmental Mathematics 16
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
17. Martin-Gay, Developmental Mathematics 17
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
18. Martin-Gay, Developmental Mathematics 18
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
19. Martin-Gay, Developmental Mathematics 19
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
20. Martin-Gay, Developmental Mathematics 20
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!
22. Martin-Gay, Developmental Mathematics 22
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.
23. Martin-Gay, Developmental Mathematics 23
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.
24. Martin-Gay, Developmental Mathematics 24
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.
25. Martin-Gay, Developmental Mathematics 25
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
26. Martin-Gay, Developmental Mathematics 26
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.
27. Martin-Gay, Developmental Mathematics 27
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.
28. Martin-Gay, Developmental Mathematics 28
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.
29. Martin-Gay, Developmental Mathematics 29
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
30. Martin-Gay, Developmental Mathematics 30
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.
31. Martin-Gay, Developmental Mathematics 31
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.
32. Martin-Gay, Developmental Mathematics 32
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
33. Martin-Gay, Developmental Mathematics 33
Factor the polynomial 6x2
y2
– 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.
6x2
y2
– 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.
34. Martin-Gay, Developmental Mathematics 34
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.
35. Martin-Gay, Developmental Mathematics 35
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.
36. Martin-Gay, Developmental Mathematics 36
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.
37. Martin-Gay, Developmental Mathematics 37
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
6x2
y2
– 2xy2
– 60y2
will be 2y2
(3x – 10)(x + 3).
Factoring Polynomials
Example Continued
39. Martin-Gay, Developmental Mathematics 39
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
40. Martin-Gay, Developmental Mathematics 40
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
42. Martin-Gay, Developmental Mathematics 42
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
44. Martin-Gay, Developmental Mathematics 44
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
45. Martin-Gay, Developmental Mathematics 45
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
46. Martin-Gay, Developmental Mathematics 46
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
47. Martin-Gay, Developmental Mathematics 47
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
48. Martin-Gay, Developmental Mathematics 48
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.
50. Martin-Gay, Developmental Mathematics 50
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.
51. Martin-Gay, Developmental Mathematics 51
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
52. Martin-Gay, Developmental Mathematics 52
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.
53. Martin-Gay, Developmental Mathematics 53
• 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
54. Martin-Gay, Developmental Mathematics 54
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
55. Martin-Gay, Developmental Mathematics 55
• 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 44
+ = − − + = − − = − −− =−
true
• So our solutions for x are or .8
1
4
5
−
Example Continued
Solving Quadratic Equations
56. Martin-Gay, Developmental Mathematics 56
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
57. Martin-Gay, Developmental Mathematics 57
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
59. Martin-Gay, Developmental Mathematics 59
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
1) Translate the problem into an equation
2) Solve the equation
3) Interpret the result
• Check proposed solution in problem
• State your conclusion
60. Martin-Gay, Developmental Mathematics 60
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
61. Martin-Gay, Developmental Mathematics 61
Finding an Unknown Number
Example continued
2.) Translate
Continued
two consecutive positive integers
x (x + 1)
is
=
132
132•
The product of
62. Martin-Gay, Developmental Mathematics 62
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)
63. Martin-Gay, Developmental Mathematics 63
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.
64. Martin-Gay, Developmental Mathematics 64
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
65. Martin-Gay, Developmental Mathematics 65
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 - 10x
x
66. Martin-Gay, Developmental Mathematics 66
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)
67. Martin-Gay, Developmental Mathematics 67
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.)