Rational expressions are expressions of the form P/Q, where P and Q are polynomials. Polynomials are expressions of the form anxn + an-1xn-1 + ... + a1x1 + a0. Rational expressions can be written in either expanded or factored form. The factored form is useful for determining the domain of a rational expression, solving equations involving rational expressions, evaluating inputs, and determining the sign of outputs. The domain of a rational expression excludes values of x that make the denominator equal to 0.
Mat221 5.6 definite integral substitutions and the area between two curvesGlenSchlee
This document contains examples of evaluating definite integrals using substitution and finding the area between curves. It includes 8 examples of evaluating definite integrals using techniques like u-substitution, integration by parts, and trigonometric substitutions. It also contains 3 examples of finding the area of regions bounded by graphs by setting up and evaluating definite integrals with respect to x or y.
To solve this rational equation, we first find the LCD of the fractions, which is x^2(x-5)(x-6). Then we multiply both sides of the equation by the LCD:
x^2(x-5)(x-6) = x^2(x-5)(x-6)
Canceling the common factors, we obtain:
x^2 = x^2
Since this holds for all real values of x, the solution is all real numbers. Therefore, the solution is x ∈ R.
GR 8 Math Powerpoint about Polynomial Techniquesreginaatin
-This is a powerpoint inspired by one of Canva displayed presentation.
- This is about Math Polynomials and good for highschoolers presentation for school.
- It consists of 39 pages explaining each of the Polynomial Techniques.
- Good for review or inspired powerpoint.
This is a PowerPoint on teaching the subject of Polynomials, Monomials, and Rational Expressions, dealing with how to add, subtract, multiply, and simplify all of these.
13 multiplication and division of rational expressionselem-alg-sample
The document discusses multiplication and division of rational expressions. It presents the multiplication rule for rational expressions as the product of the numerators over the product of the denominators. It provides an example of simplifying a rational expression by factoring the top and bottom and canceling like terms. It then gives another example with two parts, simplifying and expanding the answers of rational expression operations.
This module introduces polynomial functions of degree greater than 2. It covers identifying polynomial functions from relations, determining the degree of a polynomial, finding quotients of polynomials using division algorithm and synthetic division, and applying the remainder and factor theorems. The document provides examples and practice problems for each topic. It aims to teach students how to work with higher degree polynomial functions.
The document discusses rational expressions, which are expressions of the form P/Q where P and Q are polynomials. Polynomials are expressions involving powers of variables with numerical coefficients. Rational expressions include polynomials as a special case where P is viewed as P/1. They may be written in expanded or factored form. The factored form is useful for determining the domain of a rational expression, solving equations involving rational expressions, evaluating expressions for given inputs, and determining the signs of outputs. The domain excludes values of x that make the denominator equal to 0.
This document discusses algebraic expressions and polynomial expressions. It provides examples of algebraic expressions and defines them as formulas constructed with variables and numbers using basic arithmetic operations. Polynomials are defined as expressions of the form anxn + an-1xn-1 + ... + a1x + a0, where the ai's are numbers. The document gives examples of factoring polynomials and evaluating polynomial expressions at given values. It also discusses using factoring to find the roots of polynomial equations.
Mat221 5.6 definite integral substitutions and the area between two curvesGlenSchlee
This document contains examples of evaluating definite integrals using substitution and finding the area between curves. It includes 8 examples of evaluating definite integrals using techniques like u-substitution, integration by parts, and trigonometric substitutions. It also contains 3 examples of finding the area of regions bounded by graphs by setting up and evaluating definite integrals with respect to x or y.
To solve this rational equation, we first find the LCD of the fractions, which is x^2(x-5)(x-6). Then we multiply both sides of the equation by the LCD:
x^2(x-5)(x-6) = x^2(x-5)(x-6)
Canceling the common factors, we obtain:
x^2 = x^2
Since this holds for all real values of x, the solution is all real numbers. Therefore, the solution is x ∈ R.
GR 8 Math Powerpoint about Polynomial Techniquesreginaatin
-This is a powerpoint inspired by one of Canva displayed presentation.
- This is about Math Polynomials and good for highschoolers presentation for school.
- It consists of 39 pages explaining each of the Polynomial Techniques.
- Good for review or inspired powerpoint.
This is a PowerPoint on teaching the subject of Polynomials, Monomials, and Rational Expressions, dealing with how to add, subtract, multiply, and simplify all of these.
13 multiplication and division of rational expressionselem-alg-sample
The document discusses multiplication and division of rational expressions. It presents the multiplication rule for rational expressions as the product of the numerators over the product of the denominators. It provides an example of simplifying a rational expression by factoring the top and bottom and canceling like terms. It then gives another example with two parts, simplifying and expanding the answers of rational expression operations.
This module introduces polynomial functions of degree greater than 2. It covers identifying polynomial functions from relations, determining the degree of a polynomial, finding quotients of polynomials using division algorithm and synthetic division, and applying the remainder and factor theorems. The document provides examples and practice problems for each topic. It aims to teach students how to work with higher degree polynomial functions.
The document discusses rational expressions, which are expressions of the form P/Q where P and Q are polynomials. Polynomials are expressions involving powers of variables with numerical coefficients. Rational expressions include polynomials as a special case where P is viewed as P/1. They may be written in expanded or factored form. The factored form is useful for determining the domain of a rational expression, solving equations involving rational expressions, evaluating expressions for given inputs, and determining the signs of outputs. The domain excludes values of x that make the denominator equal to 0.
This document discusses algebraic expressions and polynomial expressions. It provides examples of algebraic expressions and defines them as formulas constructed with variables and numbers using basic arithmetic operations. Polynomials are defined as expressions of the form anxn + an-1xn-1 + ... + a1x + a0, where the ai's are numbers. The document gives examples of factoring polynomials and evaluating polynomial expressions at given values. It also discusses using factoring to find the roots of polynomial equations.
Real numbers include rational numbers like fractions and irrational numbers like square roots. Real numbers are represented by the symbol R. They consist of natural numbers, whole numbers, integers, rational numbers and irrational numbers. [/SUMMARY]
The document discusses terms, factors, and cancellation in mathematics expressions. It defines a term as one or more quantities that are added or subtracted, and a factor as a quantity that is multiplied to other quantities. Cancellation can be used to simplify fractions by canceling common factors in the numerator and denominator. However, terms cannot be canceled as they represent distinct quantities being added or subtracted. Several examples demonstrate identifying terms and factors and applying cancellation when possible.
The document discusses techniques for combining fractions with opposite denominators. It explains that we can multiply the numerator and denominator by -1 to change the denominator to its opposite. It provides examples of switching fractions to their opposite denominators and combining fractions with opposite denominators by first switching one denominator so they are the same. It also discusses an alternative approach of pulling out a "-" from the denominator and passing it to the numerator when switching denominators, ensuring the leading term is positive for polynomial denominators.
This module discusses methods for finding the zeros of polynomial functions of degree greater than 2, including: factor theorem, factoring, synthetic division, and depressed equations. It introduces the number of roots theorem, which states that a polynomial of degree n has n roots. It also discusses determining the rational zeros of a polynomial using the rational roots theorem and factor theorem. Examples are provided to illustrate these concepts and methods.
The document discusses higher order derivatives. It defines the nth derivative of a function f(x) as f(n)(x). The first example finds the first five derivatives of f(x)=2x^4 - x^3 - 2. The second example finds the first three derivatives of f(x)=-x^2/3. The third example finds the first four derivatives of f(x)=ln(x) and discusses how derivatives of rational functions become more complicated with higher orders. It also provides examples of finding derivatives of other functions like sin(x).
This document discusses various methods for finding the zeros or roots of polynomial functions, including factoring, factor theorem, synthetic division, and using the principle that every polynomial of degree n has n zeros. It provides examples of finding the zeros of polynomials by factorization, using a given zero to find other zeros through synthetic division, and identifying which numbers are zeros of various polynomials. Exercises are included for students to practice finding remaining zeros given one zero and identifying polynomial factors.
Here are the remainders when dividing the given polynomials by the specified polynomials:
1. The remainder is 0. Therefore, x-1 is a factor of x3+3x2-4x+2.
2. The remainder is 5.
3. The remainder is 0. Therefore, x+2 is a factor of 2x3+5x2+3x+11.
4. The remainder is 4.
5. The remainder is 7.
6. The remainder is 2.
Factoring is writing a polynomial as a product of two or more polynomials. The main techniques for factoring polynomials are finding the greatest common factor, factoring trinomials of the form ax^2 + bx + c, using special factoring patterns like the difference and sum of squares, and factoring polynomials with four or more terms by grouping. The goal is to factor the polynomial completely into prime factors that cannot be further factored.
This document 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 discusses techniques for finding the zeros of polynomial functions, including:
- Using the Fundamental Theorem of Algebra to determine the number of zeros
- Finding rational and complex zeros through techniques like the Rational Zero Test
- Factoring polynomials to reveal their zeros
- Applying rules like Descartes' Rule of Signs to determine possible real zeros
The document provides examples demonstrating how to apply these techniques to find all zeros of polynomial functions.
This document discusses factorials and the binomial theorem. It begins by defining factorials and providing examples of simplifying expressions with factorials. It then explains the binomial theorem, which gives a formula for expanding binomial expressions as binomial series. Specifically, it shows that the coefficients of terms in the binomial expansion can be determined using Pascal's triangle and factorials. It provides examples of using the binomial theorem to expand binomial expressions and find specific terms. In the examples, it demonstrates expanding binomials, finding coefficients, and determining terms with given exponents.
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.
The document discusses rational expressions, which are expressions of the form P/Q, where P and Q are polynomials. Polynomials are expressions involving powers of variables with numerical coefficients. Rational expressions include polynomials as a special case where P is viewed as P/1. They can be written in expanded or factored form. The factored form is useful for solving equations, determining the domain of a rational expression, evaluating inputs, and determining the sign of outputs. The domain excludes values that make the denominator equal to 0.
53 multiplication and division of rational expressionsalg1testreview
The document discusses multiplication and division of rational expressions. It presents the multiplication rule for rational expressions, which states that the product of two rational expressions is equal to the product of the tops divided by the product of the bottoms. Examples are provided to demonstrate simplifying rational expression products by factoring and canceling like terms.
This document provides instructions on how to multiply, divide, and factor polynomials. It discusses:
1) Multiplying polynomials by distributing terms and using FOIL for binomials.
2) Dividing polynomials using long division.
3) Factoring polynomials using grouping, finding two numbers whose product is the constant and sum is the coefficient, and recognizing difference of squares.
The document discusses solving literal equations by isolating the variable of interest on one side of the equation. It provides examples of solving equations for various variables by adding, subtracting, multiplying, or dividing both sides of the equation by the same quantity. The goal is to isolate the variable being solved for so it stands alone on one side of the equal sign. Steps include clearing fractions, moving all other terms to the other side of the equation, and then dividing both sides by the coefficient of the variable being solved for.
The document discusses polynomials and polynomial functions. It defines a polynomial as a sum of monomials, with a monomial being a variable or the product of a variable and real numbers with whole number exponents. It classifies polynomials by degree and number of terms, with examples of common types like linear, quadratic, and cubic polynomials. It also defines a polynomial function as a function represented by a polynomial, and discusses finding sums, differences, and writing polynomials in standard form.
This document provides an overview of polynomials, including:
- Defining polynomials as expressions involving variables and coefficients using addition, subtraction, multiplication, and exponents.
- Discussing the history of polynomial notation pioneered by Descartes.
- Explaining the different types of polynomials like monomials, binomials, and trinomials.
- Outlining common uses of polynomials in mathematics, science, and other fields.
- Describing how to find the degree of a polynomial and graph polynomial functions.
- Explaining arithmetic operations like addition, subtraction, and division that can be performed on polynomials.
A polynomial is an expression involving terms with variables that are raised to nonnegative integer powers. Polynomials are usually written in standard form by placing terms in descending order of degree. The degree of a polynomial is the highest degree of its terms. Polynomials can be added or subtracted by collecting like terms.
The document discusses sign charts and how to determine the signs of outputs for polynomials and rational expressions. It provides examples of factoring polynomials to determine if the output is positive or negative for given values of x. The key steps to create a sign chart are: 1) solve for f=0 and any undefined values, 2) mark these values on a number line, 3) sample points in each segment to determine the sign in that region. Sign charts indicate the regions where a function is positive, negative or zero.
The document discusses using sign charts to solve polynomial and rational inequalities. It provides examples of solving inequalities by setting one side equal to zero, factoring the expression, drawing the sign chart, and determining the solutions from the regions with the appropriate signs. Specifically, it works through examples of solving x^2 - 3x > 4, 2x^2 - x^3/(x^2 - 2x + 1) < 0, and (x - 2)/(2/(x - 1)) < 3.
Rational Expressions
GRADE 8 - MATHEMATICS
To get/buy a soft copy, please send a request to queenyedda@gmail.com
Inclusions of the file attachment:
* Fonts used
* Soft copy of the whole ppt file with effects
* Complete activities
PRICE: P200 only
Real numbers include rational numbers like fractions and irrational numbers like square roots. Real numbers are represented by the symbol R. They consist of natural numbers, whole numbers, integers, rational numbers and irrational numbers. [/SUMMARY]
The document discusses terms, factors, and cancellation in mathematics expressions. It defines a term as one or more quantities that are added or subtracted, and a factor as a quantity that is multiplied to other quantities. Cancellation can be used to simplify fractions by canceling common factors in the numerator and denominator. However, terms cannot be canceled as they represent distinct quantities being added or subtracted. Several examples demonstrate identifying terms and factors and applying cancellation when possible.
The document discusses techniques for combining fractions with opposite denominators. It explains that we can multiply the numerator and denominator by -1 to change the denominator to its opposite. It provides examples of switching fractions to their opposite denominators and combining fractions with opposite denominators by first switching one denominator so they are the same. It also discusses an alternative approach of pulling out a "-" from the denominator and passing it to the numerator when switching denominators, ensuring the leading term is positive for polynomial denominators.
This module discusses methods for finding the zeros of polynomial functions of degree greater than 2, including: factor theorem, factoring, synthetic division, and depressed equations. It introduces the number of roots theorem, which states that a polynomial of degree n has n roots. It also discusses determining the rational zeros of a polynomial using the rational roots theorem and factor theorem. Examples are provided to illustrate these concepts and methods.
The document discusses higher order derivatives. It defines the nth derivative of a function f(x) as f(n)(x). The first example finds the first five derivatives of f(x)=2x^4 - x^3 - 2. The second example finds the first three derivatives of f(x)=-x^2/3. The third example finds the first four derivatives of f(x)=ln(x) and discusses how derivatives of rational functions become more complicated with higher orders. It also provides examples of finding derivatives of other functions like sin(x).
This document discusses various methods for finding the zeros or roots of polynomial functions, including factoring, factor theorem, synthetic division, and using the principle that every polynomial of degree n has n zeros. It provides examples of finding the zeros of polynomials by factorization, using a given zero to find other zeros through synthetic division, and identifying which numbers are zeros of various polynomials. Exercises are included for students to practice finding remaining zeros given one zero and identifying polynomial factors.
Here are the remainders when dividing the given polynomials by the specified polynomials:
1. The remainder is 0. Therefore, x-1 is a factor of x3+3x2-4x+2.
2. The remainder is 5.
3. The remainder is 0. Therefore, x+2 is a factor of 2x3+5x2+3x+11.
4. The remainder is 4.
5. The remainder is 7.
6. The remainder is 2.
Factoring is writing a polynomial as a product of two or more polynomials. The main techniques for factoring polynomials are finding the greatest common factor, factoring trinomials of the form ax^2 + bx + c, using special factoring patterns like the difference and sum of squares, and factoring polynomials with four or more terms by grouping. The goal is to factor the polynomial completely into prime factors that cannot be further factored.
This document 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 discusses techniques for finding the zeros of polynomial functions, including:
- Using the Fundamental Theorem of Algebra to determine the number of zeros
- Finding rational and complex zeros through techniques like the Rational Zero Test
- Factoring polynomials to reveal their zeros
- Applying rules like Descartes' Rule of Signs to determine possible real zeros
The document provides examples demonstrating how to apply these techniques to find all zeros of polynomial functions.
This document discusses factorials and the binomial theorem. It begins by defining factorials and providing examples of simplifying expressions with factorials. It then explains the binomial theorem, which gives a formula for expanding binomial expressions as binomial series. Specifically, it shows that the coefficients of terms in the binomial expansion can be determined using Pascal's triangle and factorials. It provides examples of using the binomial theorem to expand binomial expressions and find specific terms. In the examples, it demonstrates expanding binomials, finding coefficients, and determining terms with given exponents.
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.
The document discusses rational expressions, which are expressions of the form P/Q, where P and Q are polynomials. Polynomials are expressions involving powers of variables with numerical coefficients. Rational expressions include polynomials as a special case where P is viewed as P/1. They can be written in expanded or factored form. The factored form is useful for solving equations, determining the domain of a rational expression, evaluating inputs, and determining the sign of outputs. The domain excludes values that make the denominator equal to 0.
53 multiplication and division of rational expressionsalg1testreview
The document discusses multiplication and division of rational expressions. It presents the multiplication rule for rational expressions, which states that the product of two rational expressions is equal to the product of the tops divided by the product of the bottoms. Examples are provided to demonstrate simplifying rational expression products by factoring and canceling like terms.
This document provides instructions on how to multiply, divide, and factor polynomials. It discusses:
1) Multiplying polynomials by distributing terms and using FOIL for binomials.
2) Dividing polynomials using long division.
3) Factoring polynomials using grouping, finding two numbers whose product is the constant and sum is the coefficient, and recognizing difference of squares.
The document discusses solving literal equations by isolating the variable of interest on one side of the equation. It provides examples of solving equations for various variables by adding, subtracting, multiplying, or dividing both sides of the equation by the same quantity. The goal is to isolate the variable being solved for so it stands alone on one side of the equal sign. Steps include clearing fractions, moving all other terms to the other side of the equation, and then dividing both sides by the coefficient of the variable being solved for.
The document discusses polynomials and polynomial functions. It defines a polynomial as a sum of monomials, with a monomial being a variable or the product of a variable and real numbers with whole number exponents. It classifies polynomials by degree and number of terms, with examples of common types like linear, quadratic, and cubic polynomials. It also defines a polynomial function as a function represented by a polynomial, and discusses finding sums, differences, and writing polynomials in standard form.
This document provides an overview of polynomials, including:
- Defining polynomials as expressions involving variables and coefficients using addition, subtraction, multiplication, and exponents.
- Discussing the history of polynomial notation pioneered by Descartes.
- Explaining the different types of polynomials like monomials, binomials, and trinomials.
- Outlining common uses of polynomials in mathematics, science, and other fields.
- Describing how to find the degree of a polynomial and graph polynomial functions.
- Explaining arithmetic operations like addition, subtraction, and division that can be performed on polynomials.
A polynomial is an expression involving terms with variables that are raised to nonnegative integer powers. Polynomials are usually written in standard form by placing terms in descending order of degree. The degree of a polynomial is the highest degree of its terms. Polynomials can be added or subtracted by collecting like terms.
The document discusses sign charts and how to determine the signs of outputs for polynomials and rational expressions. It provides examples of factoring polynomials to determine if the output is positive or negative for given values of x. The key steps to create a sign chart are: 1) solve for f=0 and any undefined values, 2) mark these values on a number line, 3) sample points in each segment to determine the sign in that region. Sign charts indicate the regions where a function is positive, negative or zero.
The document discusses using sign charts to solve polynomial and rational inequalities. It provides examples of solving inequalities by setting one side equal to zero, factoring the expression, drawing the sign chart, and determining the solutions from the regions with the appropriate signs. Specifically, it works through examples of solving x^2 - 3x > 4, 2x^2 - x^3/(x^2 - 2x + 1) < 0, and (x - 2)/(2/(x - 1)) < 3.
Rational Expressions
GRADE 8 - MATHEMATICS
To get/buy a soft copy, please send a request to queenyedda@gmail.com
Inclusions of the file attachment:
* Fonts used
* Soft copy of the whole ppt file with effects
* Complete activities
PRICE: P200 only
42 sign charts of factorable expressions and inequalitiesmath126
The document discusses using the factor form of expressions to determine the sign (positive or negative) of outputs. It explains that for a factorable expression f, its factor form can be used to infer if the output is positive or negative. Polynomial and rational expressions are given as examples. The document then demonstrates this process on some examples, factoring expressions and evaluating their signs for given values. It introduces the concept of a sign chart, which uses the factor form to graphically depict the positive and negative regions of a function.
The document discusses several topics in algebra including:
1. Indices laws including am x an = am + n, am ÷ an = am - n, and (am)n = amn. Negative and fractional indices are also discussed.
2. Logarithms including the definition that logarithm of 'x' to base 'a' is the power to which 'a' must be raised to give 'x'. Change of base formula is also provided.
3. Series including the definition of finite and infinite series. Notation of sigma notation ∑ is introduced to represent the sum of terms.
This document provides information about polynomials including definitions, types, terms, and relationships between coefficients and zeros. It begins with acknowledging those who helped create the presentation. It then defines a polynomial as an expression with variable terms raised to whole number powers. The main types discussed are linear, quadratic, and cubic polynomials. Linear polynomials have one zero while quadratics have two zeros and cubics have three. Relationships are defined between the zeros and coefficients. Graphs of linear and quadratic polynomials are presented. The division algorithm for polynomials is also explained.
The document discusses sign charts for factorable formulas. It provides examples of determining the sign (positive or negative) of expressions when evaluated at given values of x by factoring the expressions into their factored forms. The key steps to create a sign chart are: 1) solve for values where the expression is equal to 0, 2) mark these values on a number line, 3) select points in each segment to test the sign, 4) indicate the sign (positive or negative) in each segment based on the testing. Sign charts show the regions where an expression is positive, negative or equal to 0.
The document discusses polynomial expressions in mathematics. It defines a polynomial as the sum of monomial terms, where a monomial is a number multiplied by a variable raised to a non-negative integer power. The degree of a polynomial is defined as the highest exponent among its monomial terms. Several examples of evaluating monomials and polynomials are provided by substituting specific values for variables.
The document discusses polynomial expressions in mathematics. It defines a polynomial as the sum of monomial terms, where a monomial is a number multiplied by a variable raised to a non-negative integer power. The degree of a polynomial is defined as the highest exponent among its monomial terms. Several examples of evaluating monomials and polynomials are provided by substituting specific values for variables.
The document discusses sign charts of factorable polynomials and rational expressions. It defines a factorable polynomial as one that can be written as the product of real linear factors. An example polynomial is fully factored. Roots of the polynomial are defined as the values making each linear factor equal to zero. The order of a root is defined as the power of the corresponding factor. The Even/Odd-Order Sign Rule is stated: for a factorable polynomial, signs are the same on both sides of an even-ordered root and different on both sides of an odd-ordered root. An example sign chart is constructed applying this rule.
The document discusses sign charts of factorable polynomials. A polynomial is factorable if it can be written as the product of linear factors. The sign chart of a factorable polynomial follows an important rule: if a root has an even order, the signs are the same on both sides; if a root has an odd order, the signs are different on both sides. This is called the even/odd-order sign rule. An example demonstrates finding the sign chart of a polynomial by identifying the roots and their orders, and then applying the sign rule.
Rational Zeros and Decarte's Rule of Signsswartzje
The Rational Zero Theorem provides a method to determine all possible rational zeros of a polynomial function. It states that if p/q is a rational zero, then p is a factor of the constant term and q is a factor of the leading coefficient. Descartes' Rule of Signs can be used to determine the maximum number of positive and negative real zeros by counting the variations in sign of the polynomial function and its substitution of -x. It provides bounds on the number of positive and negative real zeros that are either the number of variations in sign or less by an even integer. The example demonstrates applying these methods to determine all 16 possible rational zeros and the bounds of 0 positive and either 3 or 1 negative real zeros for the given polynomial.
Project in math BY:Samuel Vasquez Baliasamuel balia
Real numbers include rational numbers like fractions as well as irrational numbers like the square root of 2. Real numbers are represented by the symbol R and include natural numbers, whole numbers, integers, rational numbers, and irrational numbers. Rational numbers can be written as fractions with integers as the numerator and non-zero denominator, while irrational numbers cannot be expressed as fractions.
The document defines and describes various types of number systems. It discusses natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers. It also describes their properties and relationships. Different types of polynomials are defined based on their degree, number of terms, zeros, and factors. Methods for factorizing polynomials including taking common factors, grouping, and splitting the middle term are explained. Algebraic identities are also introduced.
This document defines and explains polynomials. It begins by defining a polynomial as an algebraic expression with variables that have only non-negative integer powers. It then discusses the degree of polynomials with one and two variables. The document provides examples of classifying polynomials by degree and number of terms. It also covers standard form, the remainder theorem, factor theorem, and algebraic identities for factorizing polynomials. It concludes with questions and answers that apply these concepts.
This document defines polynomials and discusses their key properties. It begins by defining a polynomial as an algebraic expression with two or more terms where the power of each variable is a positive integer. The degree of a polynomial is defined as the highest power of the variable. Polynomials are then classified based on their degree as constant, linear, quadratic, cubic, etc. The document also discusses the zeros or roots of a polynomial, which are the values that make the polynomial equal to zero. It shows how the zeros relate to the coefficients of the polynomial and can be found using factoring or solving techniques. Examples are provided to illustrate dividing polynomials using the division algorithm.
1.2 review on algebra 2-sign charts and inequalitiesmath265
The document discusses sign charts and inequalities. It explains that sign charts can be used to determine if expressions are positive or negative by factoring them and evaluating at given values of x. Examples are provided to demonstrate how to construct a sign chart by: 1) solving for where the expression equals 0, 2) marking these values on a number line, and 3) evaluating the expression at sample points in each segment to determine the signs in between values where the expression equals 0. The sign chart then indicates the ranges where the expression is positive, negative or zero.
This PowerPoint presentation covers polynomials, including:
- Definitions of polynomials, monomials, binomials, trinomials, and the degree of a polynomial.
- The geometric meaning of zeros of polynomials - linear polynomials have one zero, quadratics have up to two zeros, and cubics have up to three.
- The relationship between the zeros and coefficients of a quadratic polynomial - the sum of the zeros equals the negative of the coefficient of x divided by the coefficient of x^2, and the product of the zeros equals the constant term divided by the coefficient of x^2.
- The division algorithm for polynomials - any polynomial p(x) can be divided by a non-zero polynomial
Polynomials are mathematical expressions constructed from variables and constants using addition, subtraction, multiplication, and exponents of whole numbers. They appear in many areas of mathematics and science. Polynomials can be used to form equations that model problems in various domains. They also define polynomial functions that are used in fields like physics, chemistry, economics, and social sciences. Polynomials are classified based on their degree, with linear polynomials having degree 1 and quadratic polynomials degree 2. The maximum number of zeroes a polynomial can have is equal to its degree.
The document discusses polynomial expressions. A polynomial is the sum of monomial terms, where a monomial is a number multiplied by one or more variables raised to a non-negative integer power. Examples show evaluating polynomials by substituting values for variables and calculating each monomial term separately before combining them. A term refers to each monomial in a polynomial. Terms are identified by their variable part, such as the x2-term, x-term, or constant term.
The document discusses polynomial expressions. A polynomial is the sum of monomial terms, where a monomial is a number multiplied by one or more variables raised to a non-negative integer power. Examples show evaluating polynomials by substituting values for variables and calculating each monomial term separately before combining them. A term refers to each monomial within a polynomial. Terms are identified by their variable part, such as the x2-term, x-term, or constant term.
The document discusses equations of lines. It separates lines into two cases - horizontal/vertical lines which have slope 0 or undefined slope, and their equations are y=c or x=c; and tilted lines, whose equations can be found using the point-slope formula y-y1=m(x-x1) where m is the slope and (x1,y1) is a point on the line. It provides examples of finding equations of lines given their properties like slope and intercept points.
The document discusses the concept of slope of a line. It defines slope as the ratio of the "rise" over the "run" between two points on a line. Specifically:
- The slope of a line is calculated as the change in the y-values (rise) divided by the change in the x-values (run) between two points on the line.
- This formula is easy to memorize and captures the geometric meaning of slope as the tilt of the line.
- An example problem demonstrates calculating the slope of a line between two points by finding the difference in their x- and y-values.
The document describes the rectangular coordinate system. It establishes that a coordinate system assigns positions in a plane using ordered pairs of numbers (x,y). It defines the x-axis, y-axis, and origin at their intersection. Any point is addressed by its coordinates (x,y) where x represents horizontal distance from the origin and y represents vertical distance. The four quadrants divided by the axes are also defined based on positive and negative coordinate values. Reflections of points across the axes and origin are discussed. Finally, it introduces the concept of graphing mathematical relations between x and y coordinates to represent collections of points.
The document describes the rectangular coordinate system. It defines the system as using a grid with two perpendicular axes (x and y) that intersect at the origin (0,0). Any point in the plane can be located using its coordinates (x,y), where x is the distance from the y-axis and y is the distance from the x-axis. The four quadrants (I, II, III, IV) are defined by the intersection of the positive and negative sides of the x and y axes. Examples are given of labeling points and finding coordinates on the grid.
2 the real line, inequalities and comparative phraseselem-alg-sample
The document discusses inequalities and the real number line. It explains that real numbers are associated with positions on a line, with positive numbers to the right of zero and negative numbers to the left. An inequality relates the position of two numbers on the real number line, with the number farther to the right said to be greater than the number on the left. Examples are provided of drawing intervals on the number line and solving simple inequalities algebraically. Properties of inequalities like adding the same quantity to both sides preserving the inequality sign are also outlined.
Geometry is the study of shapes, their properties and relationships. Some basic geometric shapes include lines, rays, angles, triangles, quadrilaterals, polygons, circles and three-dimensional shapes like spheres and cubes. Formulas are used to calculate properties of shapes like the area of a triangle is 1/2 * base * height, the circumference of a circle is 2 * pi * radius, and the volume of a cube is side^3.
The document discusses direct and inverse variations. It defines a direct variation as a relationship where y=kx, where k is a constant. An inverse variation is defined as a relationship where y=k/x, where k is a constant. Examples are given of translating phrases describing direct and inverse variations into mathematical equations. The document also explains how to solve word problems involving variations by using given values to find the specific constant k and exact variation equation.
17 applications of proportions and the rational equationselem-alg-sample
The document discusses rational equations word problems involving rates, distances, costs, and number of people. An example problem asks how many people (x) shared a taxi costing $20 if one person leaving causes the remaining people's cost to increase by $1 each. Setting up rational equations and solving leads to the answer that x = 5 people.
16 the multiplier method for simplifying complex fractionselem-alg-sample
The document discusses two methods for simplifying complex fractions. A complex fraction is a fraction with fractions in the numerator or denominator. The first method reduces the complex fraction to an "easy" regular division problem by combining fractions in the numerator and denominator. The second method multiplies the lowest common denominator of all terms to the numerator and denominator to simplify. An example using each method is provided.
15 proportions and the multiplier method for solving rational equationselem-alg-sample
The document discusses addition and subtraction of rational expressions. It states that rational expressions can only be added or subtracted if they have the same denominator. It provides the rule for adding or subtracting rational expressions with the same denominator. It also discusses converting rational expressions to have a common denominator so they can be added or subtracted, using the least common multiple of the denominators. Examples are provided to demonstrate converting rational expressions to equivalent forms with different specified denominators.
14 the lcm and the multiplier method for addition and subtraction of rational...elem-alg-sample
The document discusses methods for finding the least common multiple (LCM) of numbers. It defines a multiple as a number that can be divided evenly by another number. The LCM is the smallest number that is a multiple of all numbers given. Two methods are described: the searching method which tests multiples of the largest number, and the construction method which factors each number and multiplies the highest powers of common factors. Examples are provided to illustrate both methods.
The document discusses applications of factoring polynomials. It provides examples of how factoring can be used to evaluate polynomials by substituting values into the factored form. Factoring is also useful for determining the sign of outputs and for solving polynomial equations, which is described as the most important application of factoring. Examples are given to demonstrate evaluating polynomials both with and without factoring, and checking the answers obtained from factoring using the expanded form.
10 more on factoring trinomials and factoring by formulaselem-alg-sample
The document discusses two methods for factoring trinomials of the form ax^2 + bx + c. The first method is short but not always reliable, while the second method takes more steps but always provides a definite answer. This second method, called the reversed FOIL method, involves finding four numbers that satisfy certain properties to factor the trinomial. An example is worked out step-by-step to demonstrate how to use the reversed FOIL method to factor the trinomial 3x^2 + 5x + 2.
Trinomials are polynomials of the form ax^2 + bx + c, where a, b, and c are numbers. To factor a trinomial, we write it as the product of two binomials (x + u)(x + v) where uv = c and u + v = b. For example, to factor x^2 + 5x + 6, we set uv = 6 and u + v = 5. The only possible values are u = 2 and v = 3, so x^2 + 5x + 6 = (x + 2)(x + 3). Similarly, to factor x^2 - 5x + 6, we set uv = 6 and u + v = -5,
The document discusses factoring quantities by finding common factors. It defines factoring as rewriting a quantity as a product in a nontrivial way. A quantity is prime if it cannot be written as a product other than 1 times the quantity. To factor completely means writing each factor as a product of prime numbers. Examples show finding common factors of quantities, the greatest common factor (GCF), and extracting common factors from sums and differences using the extraction law.
The document discusses methods for multiplying binomial expressions. A binomial is a two-term polynomial of the form ax + b, while a trinomial is a three-term polynomial of the form ax^2 + bx + c. The product of two binomials results in a trinomial. The FOIL method is introduced to multiply binomials, where the Front, Outer, Inner, and Last terms of each binomial are multiplied and combined. Expanding the product of a binomial and a binomial with a leading negative sign requires distributing the negative sign first before using FOIL.
The document discusses exponents and rules for exponents. It defines exponents as representing the quantity A multiplied by itself N times, written as AN. It then presents and explains the following rules for exponents:
1) Multiplication Rule: ANAK = AN+K
2) Division Rule: AN/AK = AN-K
3) Power Rule: (AN)K = ANK
4) 0-Power Rule: A0 = 1
5) Negative Power Rule: A-K = 1/AK
It provides examples to illustrate how to apply each rule when simplifying expressions with exponents.
The document discusses solving literal equations by isolating the variable of interest on one side of the equation. It provides steps to take which include clearing fractions by multiplying both sides by the LCD, moving all terms except the variable of interest to one side of the equation, and then dividing both sides by the coefficient of the isolated variable term to solve for the variable. Examples are provided to demonstrate these steps, such as solving for x in (a + b)x = c by dividing both sides by (a + b).
The document provides examples and explanations for solving linear equations with one step. It defines a linear equation as one where both sides are linear expressions, such as 3x + 10 = 34, and not containing higher powers of x. To solve a one-step linear equation, the goal is to isolate the variable x on one side by applying the opposite operation to both sides, such as adding 3 to both sides of x - 3 = 12 to get x = 15. Worked examples are provided for solving equations of the form x ± a = b and cx = d.
The document discusses expressions in mathematics. It defines expressions as calculation procedures written with numbers, variables, and operation symbols that calculate outcomes. Expressions can be combined by collecting like terms. Linear expressions take the form of ax + b, where terms can be combined by adding or subtracting the coefficients of the same variable. The example shows combining the terms of the expression 2x - 4 + 9 - 5x.
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
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.
Walmart Business+ and Spark Good for Nonprofits.pdfTechSoup
"Learn about all the ways Walmart supports nonprofit organizations.
You will hear from Liz Willett, the Head of Nonprofits, and hear about what Walmart is doing to help nonprofits, including Walmart Business and Spark Good. Walmart Business+ is a new offer for nonprofits that offers discounts and also streamlines nonprofits order and expense tracking, saving time and money.
The webinar may also give some examples on how nonprofits can best leverage Walmart Business+.
The event will cover the following::
Walmart Business + (https://business.walmart.com/plus) is a new shopping experience for nonprofits, schools, and local business customers that connects an exclusive online shopping experience to stores. Benefits include free delivery and shipping, a 'Spend Analytics” feature, special discounts, deals and tax-exempt shopping.
Special TechSoup offer for a free 180 days membership, and up to $150 in discounts on eligible orders.
Spark Good (walmart.com/sparkgood) is a charitable platform that enables nonprofits to receive donations directly from customers and associates.
Answers about how you can do more with Walmart!"
The simplified electron and muon model, Oscillating Spacetime: The Foundation...RitikBhardwaj56
Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
हिंदी वर्णमाला पीपीटी, hindi alphabet PPT presentation, hindi varnamala PPT, Hindi Varnamala pdf, हिंदी स्वर, हिंदी व्यंजन, sikhiye hindi varnmala, dr. mulla adam ali, hindi language and literature, hindi alphabet with drawing, hindi alphabet pdf, hindi varnamala for childrens, hindi language, hindi varnamala practice for kids, https://www.drmullaadamali.com
Executive Directors Chat Leveraging AI for Diversity, Equity, and InclusionTechSoup
Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
3. Rational Expressions
For example,
7x12 – 4x5 + 6x2 – 5,
Polynomials are expressions of the form
anxn + an-1xn-1+ an-2xn-2+…+ a1x1+ a0
where the a’s are numbers.
4. Rational Expressions
For example,
7x12 – 4x5 + 6x2 – 5, πx – 0.762 are polynomials.
Polynomials are expressions of the form
anxn + an-1xn-1+ an-2xn-2+…+ a1x1+ a0
where the a’s are numbers.
5. Rational Expressions
For example,
7x12 – 4x5 + 6x2 – 5, πx – 0.762 are polynomials.
Rational (fractional) expressions are expressions of the
form , where P and Q are polynomials.P
Q
Polynomials are expressions of the form
anxn + an-1xn-1+ an-2xn-2+…+ a1x1+ a0
where the a’s are numbers.
6. Rational Expressions
For example,
7x12 – 4x5 + 6x2 – 5, πx – 0.762 are polynomials.
Rational (fractional) expressions are expressions of the
form , where P and Q are polynomials.P
Q
Polynomials are expressions of the form
anxn + an-1xn-1+ an-2xn-2+…+ a1x1+ a0
where the a’s are numbers.
All polynomials are rational expressions by viewing P as .
P
1
7. Rational Expressions
For example,
7x12 – 4x5 + 6x2 – 5, πx – 0.762 are polynomials.
Rational (fractional) expressions are expressions of the
form , where P and Q are polynomials.P
Q
Polynomials are expressions of the form
anxn + an-1xn-1+ an-2xn-2+…+ a1x1+ a0
where the a’s are numbers.
All polynomials are rational expressions by viewing P as .
P
1
x – 2
x2 – 2 x + 1 ,
8. Rational Expressions
For example,
7x12 – 4x5 + 6x2 – 5, πx – 0.762 are polynomials.
Rational (fractional) expressions are expressions of the
form , where P and Q are polynomials.P
Q
Polynomials are expressions of the form
anxn + an-1xn-1+ an-2xn-2+…+ a1x1+ a0
where the a’s are numbers.
All polynomials are rational expressions by viewing P as .
P
1
x – 2
x2 – 2 x + 1 ,
x(x – 2)
(x + 1) (2x + 1) are rational expressions.
9. Rational Expressions
For example,
7x12 – 4x5 + 6x2 – 5, πx – 0.762 are polynomials.
Rational (fractional) expressions are expressions of the
form , where P and Q are polynomials.P
Q
Polynomials are expressions of the form
anxn + an-1xn-1+ an-2xn-2+…+ a1x1+ a0
where the a’s are numbers.
All polynomials are rational expressions by viewing P as .
P
1
x – 2
x2 – 2 x + 1 ,
x(x – 2)
(x + 1) (2x + 1) are rational expressions.
x – 2
2 x + 1
is not a rational expression because the
denominator is not a polynomial.
10. Rational Expressions
For example,
7x12 – 4x5 + 6x2 – 5, πx – 0.762 are polynomials.
Rational (fractional) expressions are expressions of the
form , where P and Q are polynomials.P
Q
Polynomials are expressions of the form
anxn + an-1xn-1+ an-2xn-2+…+ a1x1+ a0
where the a’s are numbers.
All polynomials are rational expressions by viewing P as .
P
1
x – 2
x2 – 2 x + 1 ,
x(x – 2)
(x + 1) (2x + 1) are rational expressions.
x – 2
2 x + 1
is not a rational expression because the
denominator is not a polynomial.
Rational expressions are expressions that describe
calculation procedures that involve division (of polynomials).
11. Rational Expressions
Just as polynomials may be given in the expanded form or
factored form, rational expressions may also be given in
these two forms.
12. Rational Expressions
Just as polynomials may be given in the expanded form or
factored form, rational expressions may also be given in
these two forms.
The rational expression
x2 – 4
x2 + 2x + 1
is in the expanded form.
13. Rational Expressions
Just as polynomials may be given in the expanded form or
factored form, rational expressions may also be given in
these two forms.
The rational expression
x2 – 4
(x + 2)(x – 2)
(x + 1)(x + 1) .
is in the expanded form.
In the factored form, it’s
x2 + 2x + 1
14. Rational Expressions
Just as polynomials may be given in the expanded form or
factored form, rational expressions may also be given in
these two forms.
The rational expression
x2 – 4
x2 + 2x + 1
(x + 2)(x – 2)
(x + 1)(x + 1) .
x – 2
x2 + 1The expression is in both forms.
is in the expanded form.
In the factored form, it’s
Example A. Put the following expressions in the factored form.
a. x2 – 3x – 10
x2 – 3x
b. x2 – 3x + 10
x2 – 3
15. Rational Expressions
Just as polynomials may be given in the expanded form or
factored form, rational expressions may also be given in
these two forms.
The rational expression
x2 – 4
x2 + 2x + 1
(x + 2)(x – 2)
(x + 1)(x + 1) .
x – 2
x2 + 1The expression is in both forms.
is in the expanded form.
Example A. Put the following expressions in the factored form.
(x – 5)(x + 2)a.
x(x – 3)
x2 – 3x – 10
x2 – 3x
=
b. x2 – 3x + 10
x2 – 3
In the factored form, it’s
16. Rational Expressions
Just as polynomials may be given in the expanded form or
factored form, rational expressions may also be given in
these two forms.
The rational expression
x2 – 4
x2 + 2x + 1
(x + 2)(x – 2)
(x + 1)(x + 1) .
x – 2
x2 + 1The expression is in both forms.
is in the expanded form.
Example A. Put the following expressions in the factored form.
(x – 5)(x + 2)a.
x(x – 3)
x2 – 3x – 10
x2 – 3x
=
b. x2 – 3x + 10
x2 – 3
is in the factored form
In the factored form, it’s
17. Rational Expressions
Just as polynomials may be given in the expanded form or
factored form, rational expressions may also be given in
these two forms.
The rational expression
x2 – 4
x2 + 2x + 1
(x + 2)(x – 2)
(x + 1)(x + 1) .
x – 2
x2 + 1The expression is in both forms.
is in the expanded form.
Example A. Put the following expressions in the factored form.
(x – 5)(x + 2)a.
x(x – 3)
x2 – 3x – 10
x2 – 3x
=
b. x2 – 3x + 10
x2 – 3
is in the factored form
Note that in b. the entire (x2 – 3x + 10) or (x2 – 3) are viewed
as a single factors because they can’t be factored further.
In the factored form, it’s
18. We use the factored form to
1. solve equations
Rational Expressions
19. We use the factored form to
1. solve equations
2. determine the domain of rational expressions
Rational Expressions
20. We use the factored form to
1. solve equations
2. determine the domain of rational expressions
2. evaluate given inputs
Rational Expressions
21. We use the factored form to
1. solve equations
2. determine the domain of rational expressions
2. evaluate given inputs
3. determine of the signs of the outputs
Rational Expressions
22. We use the factored form to
1. solve equations
2. determine the domain of rational expressions
2. evaluate given inputs
3. determine of the signs of the outputs
Solutions of Equations
Rational Expressions
23. We use the factored form to
1. solve equations
2. determine the domain of rational expressions
2. evaluate given inputs
3. determine of the signs of the outputs
Solutions of Equations
The solutions of the equation of the form = 0 are the
zeroes of the numerator, so they are the solutions of P = 0.
Rational Expressions
P
Q
24. We use the factored form to
1. solve equations
2. determine the domain of rational expressions
2. evaluate given inputs
3. determine of the signs of the outputs
Solutions of Equations
Example B. a. Write in the factored form and
solve the equation
The solutions of the equation of the form = 0 are the
zeroes of the numerator, so they are the solutions of P = 0.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
x3 + x2 – 2x
x2 + 4x + 3
= 0
P
Q
25. We use the factored form to
1. solve equations
2. determine the domain of rational expressions
2. evaluate given inputs
3. determine of the signs of the outputs
Solutions of Equations
Example B. a. Write in the factored form and
solve the equation
The solutions of the equation of the form = 0 are the
zeroes of the numerator, so they are the solutions of P = 0.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
x3 – 2x2 + 3x
x2 + 4x + 3
x3 + x2 – 2x
x2 + 4x + 3
= 0
Factor, we get
P
Q
=
26. We use the factored form to
1. solve equations
2. determine the domain of rational expressions
2. evaluate given inputs
3. determine of the signs of the outputs
Solutions of Equations
Example B. a. Write in the factored form and
solve the equation
The solutions of the equation of the form = 0 are the
zeroes of the numerator, so they are the solutions of P = 0.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
x3 – 2x2 + 3x
x2 + 4x + 3
= x(x + 2)(x – 1)
(x + 3)(x + 1)
x3 + x2 – 2x
x2 + 4x + 3
= 0
Factor, we get
P
Q
27. We use the factored form to
1. solve equations
2. determine the domain of rational expressions
2. evaluate given inputs
3. determine of the signs of the outputs
Solutions of Equations
Example B. a. Write in the factored form and
solve the equation
The solutions of the equation of the form = 0 are the
zeroes of the numerator, so they are the solutions of P = 0.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
x3 – 2x2 + 3x
x2 + 4x + 3
= x(x + 2)(x – 1)
(x + 3)(x + 1)
x3 + x2 – 2x
x2 + 4x + 3
= 0
Factor, we get
Hence for
x3 + x2 – 2x
x2 + 4x + 3
= 0, it must be that x(x + 2)(x – 1) = 0
P
Q
28. We use the factored form to
1. solve equations
2. determine the domain of rational expressions
2. evaluate given inputs
3. determine of the signs of the outputs
Solutions of Equations
Example B. a. Write in the factored form and
solve the equation
The solutions of the equation of the form = 0 are the
zeroes of the numerator, so they are the solutions of P = 0.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
x3 – 2x2 + 3x
x2 + 4x + 3
= x(x + 2)(x – 1)
(x + 3)(x + 1)
x3 + x2 – 2x
x2 + 4x + 3
= 0
Factor, we get
Hence for
x3 + x2 – 2x
x2 + 4x + 3
= 0, it must be that x(x + 2)(x – 1) = 0
or that x = 0, –2, 1.
P
Q
29. Domain
The domain of a formula is the set of all the numbers that we
may use as input values for x.
Rational Expressions
30. Domain
The domain of a formula is the set of all the numbers that we
may use as input values for x. Because the denominator of a
fraction can’t be 0, therefore for the rational formulas
the zeroes of the denominator Q can’t be used as inputs.
Rational Expressions
P
Q
31. Domain
The domain of a formula is the set of all the numbers that we
may use as input values for x. Because the denominator of a
fraction can’t be 0, therefore for the rational formulas
the zeroes of the denominator Q can’t be used as inputs.
in other words, the domain of are all the numbers except
where Q = 0.
Rational Expressions
P
Q
P
Q
32. Domain
b. Determine the domain of the formula
The domain of a formula is the set of all the numbers that we
may use as input values for x. Because the denominator of a
fraction can’t be 0, therefore for the rational formulas
the zeroes of the denominator Q can’t be used as inputs.
in other words, the domain of are all the numbers except
where Q = 0.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
P
Q
P
Q
33. Domain
b. Determine the domain of the formula
The domain of a formula is the set of all the numbers that we
may use as input values for x. Because the denominator of a
fraction can’t be 0, therefore for the rational formulas
the zeroes of the denominator Q can’t be used as inputs.
in other words, the domain of are all the numbers except
where Q = 0.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
x3 – 2x2 + 3x
x2 + 4x + 3
Factor expression first.
= x(x + 2)(x – 1)
(x + 3)(x + 1)
P
Q
P
Q
34. Domain
b. Determine the domain of the formula
The domain of a formula is the set of all the numbers that we
may use as input values for x. Because the denominator of a
fraction can’t be 0, therefore for the rational formulas
the zeroes of the denominator Q can’t be used as inputs.
in other words, the domain of are all the numbers except
where Q = 0.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
x3 – 2x2 + 3x
x2 + 4x + 3
Factor expression first.
= x(x + 2)(x – 1)
(x + 3)(x + 1)
Hence we can’t have
P
Q
P
Q
(x + 3)(x + 1) = 0
35. Domain
b. Determine the domain of the formula
The domain of a formula is the set of all the numbers that we
may use as input values for x. Because the denominator of a
fraction can’t be 0, therefore for the rational formulas
the zeroes of the denominator Q can’t be used as inputs.
in other words, the domain of are all the numbers except
where Q = 0.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
x3 – 2x2 + 3x
x2 + 4x + 3
Factor expression first.
= x(x + 2)(x – 1)
(x + 3)(x + 1)
Hence we can’t have
P
Q
P
Q
(x + 3)(x + 1) = 0
so that the domain is the set of all the numbers except
–1 and –3.
36. Evaluation
It is often easier to evaluate expressions in the factored form.
Rational Expressions
37. Evaluation
c. Evaluate if x = 7 using the factored form.
It is often easier to evaluate expressions in the factored form.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
38. Evaluation
c. Evaluate if x = 7 using the factored form.
It is often easier to evaluate expressions in the factored form.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
Using the factored form
x(x + 2)(x – 1)
(x + 3)(x + 1)
Plug in x = 7,
39. Evaluation
c. Evaluate if x = 7 using the factored form.
It is often easier to evaluate expressions in the factored form.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
Using the factored form
x(x + 2)(x – 1)
(x + 3)(x + 1)
Plug in x = 7, we get 7(9)(6)
(10)(8)
40. Evaluation
c. Evaluate if x = 7 using the factored form.
It is often easier to evaluate expressions in the factored form.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
Using the factored form
x(x + 2)(x – 1)
(x + 3)(x + 1)
Plug in x = 7, we get 7(9)(6)
(10)(8)
3
4
41. Evaluation
c. Evaluate if x = 7 using the factored form.
It is often easier to evaluate expressions in the factored form.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
Using the factored form
x(x + 2)(x – 1)
(x + 3)(x + 1)
Plug in x = 7, we get 7(9)(6)
(10)(8)
=
3
4
189
40
42. Evaluation
c. Evaluate if x = 7 using the factored form.
It is often easier to evaluate expressions in the factored form.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
Using the factored form
x(x + 2)(x – 1)
(x + 3)(x + 1)
Plug in x = 7, we get 7(9)(6)
(10)(8)
=
3
4
189
40
Signs
43. Evaluation
c. Evaluate if x = 7 using the factored form.
It is often easier to evaluate expressions in the factored form.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
Using the factored form
x(x + 2)(x – 1)
(x + 3)(x + 1)
Plug in x = 7, we get 7(9)(6)
(10)(8)
=
3
4
189
40
We use the factored form to determine the sign of an output.
Signs
44. Evaluation
c. Evaluate if x = 7 using the factored form.
It is often easier to evaluate expressions in the factored form.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
Using the factored form
x(x + 2)(x – 1)
(x + 3)(x + 1)
Plug in x = 7, we get 7(9)(6)
(10)(8)
=
3
4
189
40
We use the factored form to determine the sign of an output.
Signs
d. Determine the signs (positive or negative) of
if x = ½,– 5/2 using the factored form.
x3 + x2 – 2x
x2 + 4x + 3
45. Evaluation
c. Evaluate if x = 7 using the factored form.
It is often easier to evaluate expressions in the factored form.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
Using the factored form
x(x + 2)(x – 1)
(x + 3)(x + 1)
Plug in x = 7, we get 7(9)(6)
(10)(8)
=
3
4
189
40
We use the factored form to determine the sign of an output.
Signs
d. Determine the signs (positive or negative) of
if x = ½,– 5/2 using the factored form.
x3 + x2 – 2x
x2 + 4x + 3
Plug in x = ½ into the above factored form and check the signs
of each factor,
46. Evaluation
c. Evaluate if x = 7 using the factored form.
It is often easier to evaluate expressions in the factored form.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
Using the factored form
x(x + 2)(x – 1)
(x + 3)(x + 1)
Plug in x = 7, we get 7(9)(6)
(10)(8)
=
3
4
189
40
We use the factored form to determine the sign of an output.
Signs
d. Determine the signs (positive or negative) of
if x = ½,– 5/2 using the factored form.
x3 + x2 – 2x
x2 + 4x + 3
Plug in x = ½ into the above factored form and check the signs
of each factor, we get +( + )( – )
(+)(+)
47. Evaluation
c. Evaluate if x = 7 using the factored form.
It is often easier to evaluate expressions in the factored form.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
Using the factored form
x(x + 2)(x – 1)
(x + 3)(x + 1)
Plug in x = 7, we get 7(9)(6)
(10)(8)
=
3
4
189
40
We use the factored form to determine the sign of an output.
Signs
d. Determine the signs (positive or negative) of
if x = ½,– 5/2 using the factored form.
x3 + x2 – 2x
x2 + 4x + 3
Plug in x = ½ into the above factored form and check the signs
of each factor, we get = –, so it’s negative.+( + )( – )
(+)(+)
48. Evaluation
c. Evaluate if x = 7 using the factored form.
It is often easier to evaluate expressions in the factored form.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
Using the factored form
x(x + 2)(x – 1)
(x + 3)(x + 1)
Plug in x = 7, we get 7(9)(6)
(10)(8)
=
3
4
189
40
We use the factored form to determine the sign of an output.
Signs
d. Determine the signs (positive or negative) of
if x = ½,– 5/2 using the factored form.
x3 + x2 – 2x
x2 + 4x + 3
Plug in x = ½ into the above factored form and check the signs
of each factor, we get
Plug in x = –5/2 into the factored form we get –( – )( – )
(+)( – )
= –, so it’s negative.+( + )( – )
(+)(+)
49. Evaluation
c. Evaluate if x = 7 using the factored form.
It is often easier to evaluate expressions in the factored form.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
Using the factored form
x(x + 2)(x – 1)
(x + 3)(x + 1)
Plug in x = 7, we get 7(9)(6)
(10)(8)
=
3
4
189
40
We use the factored form to determine the sign of an output.
Signs
d. Determine the signs (positive or negative) of
if x = ½,– 5/2 using the factored form.
x3 + x2 – 2x
x2 + 4x + 3
Plug in x = ½ into the above factored form and check the signs
of each factor, we get = –, so it’s negative.+( + )( – )
(+)(+)
Plug in x = –5/2 into the factored form we get –( – )( – )
(+)( – ) = +
so that the output is positive.
51. PN
QN
= =
1
Rational Expressions
Cancellation Law: Common factor may be cancelled, i.e.
PN
QN
P
Q
A rational expression is reduced (simplified) if all common
factors are cancelled.
52. PN
QN
= =
1
Rational Expressions
Cancellation Law: Common factor may be cancelled, i.e.
PN
QN
P
Q
A rational expression is reduced (simplified) if all common
factors are cancelled.
To reduce (simplify) a rational expression, put the expression
in the factored form then cancel the common factors.
53. PN
QN
= =
1
Rational Expressions
Cancellation Law: Common factor may be cancelled, i.e.
PN
QN
P
Q
A rational expression is reduced (simplified) if all common
factors are cancelled.
To reduce (simplify) a rational expression, put the expression
in the factored form then cancel the common factors.
Example B. Reduce the following expressions.
(x – 2)(x + 3)
(x + 3)(x + 2)
a.
b.
x2 – 3x + 10
x2 – 3
54. PN
QN
= =
1
Rational Expressions
Cancellation Law: Common factor may be cancelled, i.e.
PN
QN
P
Q
A rational expression is reduced (simplified) if all common
factors are cancelled.
To reduce (simplify) a rational expression, put the expression
in the factored form then cancel the common factors.
Example B. Reduce the following expressions.
(x – 2)(x + 3)
It's already factored, proceed to cancel the common factor.
(x + 3)(x + 2)
(x – 2)(x + 3)
(x + 3)(x + 2)
a.
b.
x2 – 3x + 10
x2 – 3
55. PN
QN
= =
1
Rational Expressions
Cancellation Law: Common factor may be cancelled, i.e.
PN
QN
P
Q
A rational expression is reduced (simplified) if all common
factors are cancelled.
To reduce (simplify) a rational expression, put the expression
in the factored form then cancel the common factors.
Example B. Reduce the following expressions.
(x – 2)(x + 3)
1
=
x – 2
x + 2
It's already factored, proceed to cancel the common factor.
(x + 3)(x + 2)
(x – 2)(x + 3)
(x + 3)(x + 2)
which is reduced.
a.
b.
x2 – 3x + 10
x2 – 3
56. PN
QN
= =
1
Rational Expressions
Cancellation Law: Common factor may be cancelled, i.e.
PN
QN
P
Q
A rational expression is reduced (simplified) if all common
factors are cancelled.
To reduce (simplify) a rational expression, put the expression
in the factored form then cancel the common factors.
Example B. Reduce the following expressions.
(x – 2)(x + 3)
1
=
x – 2
x + 2
It's already factored, proceed to cancel the common factor.
(x + 3)(x + 2)
(x – 2)(x + 3)
(x + 3)(x + 2)
which is reduced.
a.
b.
x2 – 3x + 10
x2 – 3
This is in the factored form.
57. PN
QN
= =
1
Rational Expressions
Cancellation Law: Common factor may be cancelled, i.e.
PN
QN
P
Q
A rational expression is reduced (simplified) if all common
factors are cancelled.
To reduce (simplify) a rational expression, put the expression
in the factored form then cancel the common factors.
Example B. Reduce the following expressions.
(x – 2)(x + 3)
1
=
x – 2
x + 2
It's already factored, proceed to cancel the common factor.
(x + 3)(x + 2)
(x – 2)(x + 3)
(x + 3)(x + 2)
which is reduced.
a.
b.
x2 – 3x + 10
x2 – 3
This is in the factored form. There are no common factors
so it’s already reduced.
62. Rational Expressions
c.
x2 – 1
x2 – 3x+ 2
x2 – 1
x2 – 3x+ 2
=
(x – 1)(x + 1)
(x – 1)(x – 2)
1
=
x + 1
x – 2
Factor then cancel
63. Rational Expressions
Only factors may be canceled.
c.
x2 – 1
x2 – 3x+ 2
x2 – 1
x2 – 3x+ 2
=
(x – 1)(x + 1)
(x – 1)(x – 2)
1
=
x + 1
x – 2
Factor then cancel
64. Rational Expressions
Only factors may be canceled. It is wrong to cancel terms
c.
x2 – 1
x2 – 3x+ 2
x2 – 1
x2 – 3x+ 2
=
(x – 1)(x + 1)
(x – 1)(x – 2)
1
=
x + 1
x – 2
Factor then cancel
65. Rational Expressions
Only factors may be canceled. It is wrong to cancel terms, i.e.
P + Q
P + R
=
P + Q
P + R
c.
x2 – 1
x2 – 3x+ 2
x2 – 1
x2 – 3x+ 2
=
(x – 1)(x + 1)
(x – 1)(x – 2)
1
=
x + 1
x – 2
Factor then cancel
66. Rational Expressions
Only factors may be canceled. It is wrong to cancel terms, i.e.
P + Q
P + R
=
P + Q
P + R
For example,
x2 + 1
x2 – 2 =
x2 + 1
x2 – 2
c.
x2 – 1
x2 – 3x+ 2
x2 – 1
x2 – 3x+ 2
=
(x – 1)(x + 1)
(x – 1)(x – 2)
1
=
x + 1
x – 2
Factor then cancel
67. Rational Expressions
Only factors may be canceled. It is wrong to cancel terms, i.e.
P + Q
P + R
=
P + Q
P + R
For example,
x2 + 1
x2 – 2 =
x2 + 1
x2 – 2
c.
x2 – 1
x2 – 3x+ 2
x2 – 1
x2 – 3x+ 2
=
(x – 1)(x + 1)
(x – 1)(x – 2)
1
=
x + 1
x – 2
Factor then cancel
Cancellation of Opposite Factors
68. Rational Expressions
Only factors may be canceled. It is wrong to cancel terms, i.e.
P + Q
P + R
=
P + Q
P + R
For example,
x2 + 1
x2 – 2 =
x2 + 1
x2 – 2
The opposite of a quantity x is the –x.
c.
x2 – 1
x2 – 3x+ 2
x2 – 1
x2 – 3x+ 2
=
(x – 1)(x + 1)
(x – 1)(x – 2)
1
=
x + 1
x – 2
Factor then cancel
Cancellation of Opposite Factors
69. Rational Expressions
Only factors may be canceled. It is wrong to cancel terms, i.e.
P + Q
P + R
=
P + Q
P + R
For example,
x2 + 1
x2 – 2 =
x2 + 1
x2 – 2
The opposite of a quantity x is the –x.
original: y -z x – y v – 4u – 2w
opposite:
c.
x2 – 1
x2 – 3x+ 2
x2 – 1
x2 – 3x+ 2
=
(x – 1)(x + 1)
(x – 1)(x – 2)
1
=
x + 1
x – 2
Factor then cancel
Cancellation of Opposite Factors
70. Rational Expressions
Only factors may be canceled. It is wrong to cancel terms, i.e.
P + Q
P + R
=
P + Q
P + R
For example,
x2 + 1
x2 – 2 =
x2 + 1
x2 – 2
The opposite of a quantity x is the –x.
original: y -z x – y v – 4u – 2w
opposite: -y
c.
x2 – 1
x2 – 3x+ 2
x2 – 1
x2 – 3x+ 2
=
(x – 1)(x + 1)
(x – 1)(x – 2)
1
=
x + 1
x – 2
Factor then cancel
Cancellation of Opposite Factors
71. Rational Expressions
Only factors may be canceled. It is wrong to cancel terms, i.e.
P + Q
P + R
=
P + Q
P + R
For example,
x2 + 1
x2 – 2 =
x2 + 1
x2 – 2
The opposite of a quantity x is the –x.
original: y -z x – y v – 4u – 2w
opposite: -y z
c.
x2 – 1
x2 – 3x+ 2
x2 – 1
x2 – 3x+ 2
=
(x – 1)(x + 1)
(x – 1)(x – 2)
1
=
x + 1
x – 2
Factor then cancel
Cancellation of Opposite Factors
72. Rational Expressions
Only factors may be canceled. It is wrong to cancel terms, i.e.
P + Q
P + R
=
P + Q
P + R
For example,
x2 + 1
x2 – 2 =
x2 + 1
x2 – 2
The opposite of a quantity x is the –x.
original: y -z x – y v – 4u – 2w
opposite: -y z -x + y or y – x
c.
x2 – 1
x2 – 3x+ 2
x2 – 1
x2 – 3x+ 2
=
(x – 1)(x + 1)
(x – 1)(x – 2)
1
=
x + 1
x – 2
Factor then cancel
Cancellation of Opposite Factors
73. Rational Expressions
Only factors may be canceled. It is wrong to cancel terms, i.e.
P + Q
P + R
=
P + Q
P + R
For example,
x2 + 1
x2 – 2 =
x2 + 1
x2 – 2
The opposite of a quantity x is the –x.
original: y -z x – y v – 4u – 2w
opposite: -y z -x + y or y – x -v + 4u + 2w or …
c.
x2 – 1
x2 – 3x+ 2
x2 – 1
x2 – 3x+ 2
=
(x – 1)(x + 1)
(x – 1)(x – 2)
1
=
x + 1
x – 2
Factor then cancel
Cancellation of Opposite Factors
74. Rational Expressions
Only factors may be canceled. It is wrong to cancel terms, i.e.
P + Q
P + R
=
P + Q
P + R
For example,
x2 + 1
x2 – 2 =
x2 + 1
x2 – 2
Cancellation of Opposite Factors
The opposite of a quantity x is the –x.
While identical factors cancel to be 1, opposite factors
cancel to be –1,
original: y -z x – y v – 4u – 2w
opposite: -y z -x + y or y – x -v + 4u + 2w or …
c.
x2 – 1
x2 – 3x+ 2
x2 – 1
x2 – 3x+ 2
=
(x – 1)(x + 1)
(x – 1)(x – 2)
1
=
x + 1
x – 2
Factor then cancel
75. Rational Expressions
Only factors may be canceled. It is wrong to cancel terms, i.e.
P + Q
P + R
=
P + Q
P + R
For example,
x2 + 1
x2 – 2 =
x2 + 1
x2 – 2
The opposite of a quantity x is the –x.
While identical factors cancel to be 1, opposite factors
cancel to be –1, in symbol,
original: y -z x – y v – 4u – 2w
opposite: -y z -x + y or y – x -v + 4u + 2w or …
x
–x = –1.
c.
x2 – 1
x2 – 3x+ 2
x2 – 1
x2 – 3x+ 2
=
(x – 1)(x + 1)
(x – 1)(x – 2)
1
=
x + 1
x – 2
Factor then cancel
Cancellation of Opposite Factors
78. b(x – y)
a(y – x)
Example C.
2y
–2y
=
-1
–1a.
b.
Rational Expressions
79. b(x – y)
a(y – x)
Example C.
2y
–2y
=
-1
–1
-1
a.
b.
Rational Expressions
80. b(x – y)
a(y – x)
Example C.
2y
–2y
=
-1
–1
-1
=
–b
a
a.
b.
Rational Expressions
81. b(x – y)
a(y – x)
Example C.
2y
–2y
=
-1
–1
-1
=
–b
a
a.
b.
Rational Expressions
(x – 2y)(x + 3y)
(x – 3y)(2y – x)
c.
82. b(x – y)
a(y – x)
Example C.
2y
–2y
=
-1
–1
-1
=
–b
a
a.
b.
Rational Expressions
(x – 2y)(x + 3y)
(x – 3y)(2y – x)
c.
-1
83. b(x – y)
a(y – x)
Example C.
2y
–2y
=
-1
–1
-1
=
–b
a
a.
b.
Rational Expressions
(x – 2y)(x + 3y)
(x – 3y)(2y – x)
c.
-1
=
–(x + 3y)
(x – 3y)
84. b(x – y)
a(y – x)
Example C.
2y
–2y
=
-1
–1
-1
=
–b
a
a.
b.
Rational Expressions
(x – 2y)(x + 3y)
(x – 3y)(2y – x)
c.
-1
=
–(x + 3y)
(x – 3y)
or
–x – 3y
x – 3y
85. b(x – y)
a(y – x)
Example C.
2y
–2y
=
-1
–1
-1
=
–b
a
a.
b.
Rational Expressions
(x – 2y)(x + 3y)
(x – 3y)(2y – x)
c.
-1
=
–(x + 3y)
(x – 3y)
or
–x – 3y
x – 3y
d.
4 – x2
x2 – x – 2
86. b(x – y)
a(y – x)
Example C.
2y
–2y
=
-1
–1
-1
=
–b
a
a.
b.
Rational Expressions
(x – 2y)(x + 3y)
(x – 3y)(2y – x)
c.
-1
=
–(x + 3y)
(x – 3y)
or
–x – 3y
x – 3y
d.
4 – x2
x2 – x – 2
=
(2 – x)(2 + x)
(x + 1)(x – 2)
87. b(x – y)
a(y – x)
Example C.
2y
–2y
=
-1
–1
-1
=
–b
a
a.
b.
Rational Expressions
(x – 2y)(x + 3y)
(x – 3y)(2y – x)
c.
-1
=
–(x + 3y)
(x – 3y)
or
–x – 3y
x – 3y
d.
4 – x2
x2 – x – 2
=
(2 – x)(2 + x)
(x + 1)(x – 2)
-1
88. b(x – y)
a(y – x)
Example C.
2y
–2y
=
-1
–1
-1
=
–b
a
a.
b.
Rational Expressions
(x – 2y)(x + 3y)
(x – 3y)(2y – x)
c.
-1
=
–(x + 3y)
(x – 3y)
or
–x – 3y
x – 3y
d.
4 – x2
x2 – x – 2
=
(2 – x)(2 + x)
(x + 1)(x – 2)
-1
=
–(2 + x)
x + 1
or
–2 – x
x + 1
89. b(x – y)
a(y – x)
Example C.
2y
–2y
=
-1
–1
-1
=
–b
a
a.
b.
Rational Expressions
(x – 2y)(x + 3y)
(x – 3y)(2y – x)
c.
-1
=
–(x + 3y)
(x – 3y)
or
–x – 3y
x – 3y
d.
4 – x2
x2 – x – 2
=
(2 – x)(2 + x)
(x + 1)(x – 2)
-1
=
–(2 + x)
x + 1
or
–2 – x
x + 1
In the case of polynomials in one variable x, if the highest
degree term is negative, we may factor out the negative sign
then factor the expressions.
90. b(x – y)
a(y – x)
Example C.
2y
–2y
=
-1
–1
-1
=
–b
a
a.
b.
Rational Expressions
(x – 2y)(x + 3y)
(x – 3y)(2y – x)
c.
-1
=
–(x + 3y)
(x – 3y)
or
–x – 3y
x – 3y
d.
4 – x2
x2 – x – 2
=
(2 – x)(2 + x)
(x + 1)(x – 2)
-1
=
–(2 + x)
x + 1
or
–2 – x
x + 1
–x2 + 4
–x2 + x + 2
Example D. Pull out the “–” first then reduce.
In the case of polynomials in one variable x, if the highest
degree term is negative, we may factor out the negative sign
then factor the expressions.
91. b(x – y)
a(y – x)
Example C.
2y
–2y
=
-1
–1
-1
=
–b
a
a.
b.
Rational Expressions
(x – 2y)(x + 3y)
(x – 3y)(2y – x)
c.
-1
=
–(x + 3y)
(x – 3y)
or
–x – 3y
x – 3y
d.
4 – x2
x2 – x – 2
=
(2 – x)(2 + x)
(x + 1)(x – 2)
-1
=
–(2 + x)
x + 1
or
–2 – x
x + 1
In the case of polynomials in one variable x, if the highest
degree term is negative, we may factor out the negative sign
then factor the expressions.
–x2 + 4
–x2 + x + 2
=
Example D. Pull out the “–” first then reduce.
–(x2 – 4)
–(x2 – x – 2)
92. b(x – y)
a(y – x)
Example C.
2y
–2y
=
-1
–1
-1
=
–b
a
a.
b.
Rational Expressions
(x – 2y)(x + 3y)
(x – 3y)(2y – x)
c.
-1
=
–(x + 3y)
(x – 3y)
or
–x – 3y
x – 3y
d.
4 – x2
x2 – x – 2
=
(2 – x)(2 + x)
(x + 1)(x – 2)
-1
=
–(2 + x)
x + 1
or
–2 – x
x + 1
–x2 + 4
–x2 + x + 2
=
(x – 2)(x + 2)
(x + 1)(x – 2)
=
Example D. Pull out the “–” first then reduce.
–(x2 – 4)
–(x2 – x – 2)
In the case of polynomials in one variable x, if the highest
degree term is negative, we may factor out the negative sign
then factor the expressions.
93. b(x – y)
a(y – x)
Example C.
2y
–2y
=
-1
–1
-1
=
–b
a
a.
b.
Rational Expressions
(x – 2y)(x + 3y)
(x – 3y)(2y – x)
c.
-1
=
–(x + 3y)
(x – 3y)
or
–x – 3y
x – 3y
d.
4 – x2
x2 – x – 2
=
(2 – x)(2 + x)
(x + 1)(x – 2)
-1
=
–(2 + x)
x + 1
or
–2 – x
x + 1
–x2 + 4
–x2 + x + 2
=
(x – 2)(x + 2)
(x + 1)(x – 2)
=
Example D. Pull out the “–” first then reduce.
–(x2 – 4)
–(x2 – x – 2)
=
x + 2
x + 1
In the case of polynomials in one variable x, if the highest
degree term is negative, we may factor out the negative sign
then factor the expressions.
94. Rational Expressions
To summarize, a rational expression is reduced (simplified)
if all common factors are cancelled.
Following are the steps for reducing a rational expression.
1. Factor the top and bottom completely.
(If present, factor the “ – ” from the leading term)
1. Cancel the common factors:
-cancel identical factors to be 1
-cancel opposite factors to be –1
95. Ex. A. Write the following expressions in factored form.
List all the distinct factors of the numerator and the
denominator of each expression.
1.
Rational Expressions
2x + 3
x + 3
2.
4x + 6
2x + 6
3.
x2 – 4
2x + 4
4.
x2 + 4
x2 + 4x
5.
x2 – 2x – 3
x2 + 4x
6.
x3 – 2x2 – 8x
x2 + 2x – 3
7. Find the zeroes and list the domain of
x2 – 2x – 3
x2 + 4x
8. Use the factored form to evaluate x2 – 2x – 3
x2 + 4x
with x = 7, ½, – ½, 1/3.
9. Determine the signs of the outputs of
x2 – 2x – 3
x2 + 4x
with x = 4, –2, 1/7, 1.23.
For problems 10, 11, and 12, answer the same questions
as problems 7, 8 and 9 with the formula .x3 – 2x2 – 8x
x2 + 2x – 3
96. Ex. B. Reduce the following expressions. If it’s already
reduced, state this. Make sure you do not cancel any terms
and make sure that you look for the opposite cancellation.
13.
Rational Expressions
2x + 3
x + 3
20.
4x + 6
2x + 3
22. 23. 24.
21.
3x – 12
x – 4
12 – 3x
x – 4
4x + 6
–2x – 3
3x + 12
x – 4
25. 4x – 6
–2x – 3
14.
x + 3
x – 3
15.
x + 3
–x – 3
16.
x + 3
x – 3
17.
x – 3
3 – x
18.
2x – 1
1 + 2x
19. 2x – 1
1 – 2x
26. (2x – y)(x – 2y)
(2y + x)(y – 2x)
27. (3y + x)(3x –y)
(y – 3x)(–x – 3y)
28. (2u + v – w)(2v – u – 2w)
(u – 2v + 2w)(–2u – v – w)
29.(a + 4b – c)(a – b – c)
(c – a – 4b)(a + b + c)
97. 30.
Rational Expressions
37.
x2 – 1
x2 + 2x – 3
36. 38.
x – x2
39.
x2 – 3x – 4
31. 32.
33. 34. 35.
40. 41. x3 – 16x
x2 + 4
2x + 4
x2 – 4x + 4
x2 – 4
x2– 2x
x2 – 9
x2 + 4x + 3
x2 – 4
2x + 4
x2 + 3x + 2
x2 – x – 2 x2 + x – 2
x2 – x – 6
x2 – 5x + 6
x2 – x – 2
x2 + x – 2
x2 – 5x – 6
x2 + 5x – 6
x2 + 5x + 6
x3 – 8x2 – 20x
46.45. 47. 9 – x2
42. 43. 44.
x2 – 2x
9 – x2
x2 + 4x + 3
– x2 – x + 2
x3 – x2 – 6x
–1 + x2
–x2 + x + 2
x2 – x – 2
– x2 + 5x – 6
1 – x2
x2 + 5x – 6
49.48. 50.
xy – 2y + x2 – 2x
x2 – y2x3 – 100x
x2 – 4xy + x – 4y
x2 – 3xy – 4y2
Ex. C. Reduce the following expressions. If it’s already
reduced, state this. Make sure you do not cancel any terms
and make sure that you look for the opposite cancellation.