FINDING THE LEAST COMMON DENOMINATOR

Save this PDF as:

Size: px
Start display at page:

Download "FINDING THE LEAST COMMON DENOMINATOR"

Transcription

1 0 (7 18) Chapter 7 Rational Expressions GETTING MORE INVOLVED 7. Discussion. Evaluate each expression. a) One-half of 1 b) One-third of c) One-half of x d) One-half of x 7. Exploration. Let R 6 x x 0 x and H 9x 8 x x. 1 a) Find R when x and x. Find H when x and x. b) How are these values of R and H related and why In this section Building Up the Denominator Finding the Least Common Denominator 7. FINDING THE LEAST COMMON DENOMINATOR Every rational expression can be written in infinitely many equivalent forms. Because we can add or subtract only fractions with identical denominators, we must be able to change the denominator of a fraction. You have already learned how to change the denominator of a fraction by reducing. In this section you will learn the opposite of reducing, which is called building up the denominator. Building Up the Denominator To convert the fraction into an equivalent fraction with a denominator of 1, we factor 1 as 1 7. Because already has a in the denominator, multiply the numerator and denominator of by the missing factor 7 to get a denominator of 1: For rational expressions the process is the same. To convert the rational expression x into an equivalent rational expression with a denominator of x x 1, first factor x x 1: x x 1 (x )(x ) From the factorization we can see that the denominator x needs only a factor of x to have the required denominator. So multiply the numerator and denominator by the missing factor x : x x 0 x x x x x 1 E X A M P L E 1 Building up the denominator Build each rational expression into an equivalent rational expression with the indicated denominator. a) 1 b) w c) w x y 1y 8

2 7. Finding the Least Common Denominator (7 19) 1 a) Because, we get a denominator of 1 by multiplying the numerator and 1 denominator by 1: b) Multiply the numerator and denominator by x: x w w x x w x c) To build the denominator y up to 1y 8, multiply by y : y y 8y y y 1 y 8 In the next example we must factor the original denominator before building up the denominator. E X A M P L E helpful hint Notice that reducing and building up are exactly the opposite of each other. In reducing you remove a factor that is common to the numerator and denominator, and in building up you put a common factor into the numerator and denominator. Building up the denominator Build each rational expression into an equivalent rational expression with the indicated denominator. 7 a) b) x x y 6y 6x x x 8x 1 a) Because x y (x y), we factor 6 out of 6y 6x. This will give a factor of x y in each denominator: x y (x y) 6y 6x 6(x y) (x y) To get the required denominator, we multiply the numerator and denominator by only: 7 7() x y (x y) () 1 6y 6x b) Because x 8x 1 (x )(x 6), we multiply the numerator and denominator by x 6, the missing factor: x ( x ) ( x 6) x ( x ) ( x 6) x x 1 x 8x 1 CAUTION When building up a denominator, both the numerator and the denominator must be multiplied by the appropriate expression, because that is how we build up fractions.

3 (7 0) Chapter 7 Rational Expressions Finding the Least Common Denominator We can use the idea of building up the denominator to convert two fractions with different denominators into fractions with identical denominators. For example, 6 and 1 can both be converted into fractions with a denominator of 1, since 1 6 and 1 : The smallest number that is a multiple of all of the denominators is called the least common denominator (LCD). The LCD for the denominators 6 and is 1. To find the LCD in a systematic way, we look at a complete factorization of each denominator. Consider the denominators and 0: 0 study tip Studying in a quiet place is better than studying in a noisy place. There are very few people who can listen to music or a conversation and study at the same time. Any multiple of must have three s in its factorization, and any multiple of 0 must have one as a factor. So a number with three s in its factorization will have enough to be a multiple of both and 0. The LCD must also have one and one in its factorization. We use each factor the maximum number of times it appears in either factorization. So the LCD is : 10 0 If we omitted any one of the factors in, we would not have a multiple of both and 0. That is what makes 10 the least common denominator. To find the LCD for two polynomials, we use the same strategy. Strategy for Finding the LCD for Polynomials 1. Factor each denominator completely. Use exponent notation for repeated factors.. Write the product of all of the different factors that appear in the denominators.. On each factor, use the highest power that appears on that factor in any of the denominators. E X A M P L E Finding the LCD If the given expressions were used as denominators of rational expressions, then what would be the LCD for each group of denominators a) 0, 0 b) x yz, x y z, xyz c) a a 6, a a

4 7. Finding the Least Common Denominator (7 1) a) First factor each number completely: 0 0 The highest power of is, and the highest power of is. So the LCD of 0 and0is, or 100. b) The expressions x yz, x y z, and xyz are already factored. For the LCD, use the highest power of each variable. So the LCD is x y z. c) First factor each polynomial. a a 6 (a )(a ) a a (a ) The highest power of (a ) is 1, and the highest power of (a ) is. So the LCD is (a )(a ). When adding or subtracting rational expressions, we must convert the expressions into expressions with identical denominators. To keep the computations as simple as possible, we use the least common denominator. E X A M P L E helpful hint What is the difference between LCD, GCF, CBS, and NBC The LCD for the denominators and 6 is 1. The least common denominator is greater than or equal to both numbers. The GCF for and 6 is. The greatest common factor is less than or equal to both numbers. CBS and NBC are TV networks. Find the LCD for the rational expressions, and convert each expression into an equivalent rational expression with the LCD as the denominator. a) 9 xy, b) 1xz 6 x, 1, 8x y y a) Factor each denominator completely: 9xy xy 1xz xz The LCD is xyz. Now convert each expression into an expression with this denominator. We must multiply the numerator and denominator of the first rational expression by z and the second by y: 9xy z 0z 9xy z xyz y 6y 1xz 1 xz y xyz Same denominator b) Factor each denominator completely: 6x x 8x y x y y y The LCD is x y or x y. Now convert each expression into an expression with this denominator: 6x xy 0xy 6x xy x y 1 1 y y 8x y 8x y y x y y 6x 18x y 6x x y

5 (7 ) Chapter 7 Rational Expressions E X A M P L E Find the LCD for the rational expressions x and x x x 6 and convert each into an equivalent rational expression with that denominator. First factor the denominators: x (x )(x ) x x 6 (x )(x ) The LCD is (x )(x )(x ). Now we multiply the numerator and denominator of the first rational expression by (x ) and those of the second rational expression by (x ). Because each denominator already has one factor of (x ), there is no reason to multiply by (x ). We multiply each denominator by the factors in the LCD that are missing from that denominator: x x x x 6 x(x ) (x )(x )(x ) (x ) (x )(x )(x ) x 1x (x )(x )(x ) x 6 (x )(x )(x ) Same denominator Note that in Example we multiplied the expressions in the numerators but left the denominators in factored form. The numerators are simplified because it is the numerators that must be added when we add rational expressions in Section 7.. Because we can add rational expressions with identical denominators, there is no need to multiply the denominators. WARM-UPS True or false Explain your answer. 1. To convert into an equivalent fraction with a denominator of 18, we would multiply only the denominator of by 6.. Factoring has nothing to do with finding the least common denominator.. a b 1 a b 10a b for any nonzero values of a and b.. The LCD for the denominators and is.. The LCD for the fractions 1 6 and 1 is The LCD for the denominators 6a b and ab is ab. 7. The LCD for the denominators a 1anda 1isa 1. x x 7 8. for any real number x The LCD for the rational expressions x and x is x. 10. x x for any real number x.

6 7. Finding the Least Common Denominator (7 ) 7. EXERCISES Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences. 1. What is building up the denominator. How do we build up the denominator of a rational expression. What is the least common denominator for fractions. How do you find the LCD for two polynomial denominators Build each rational expression into an equivalent rational expression with the indicated denominator. See Example x y 9. b bt ay ayz 11. 9z 1. 7yt aw 8a wz x 18 xyt 7b a 1 a 1 c 6 c 8 1. x y y x y 8 x z x z Build each rational expression into an equivalent rational expression with the indicated denominator. See Example. 17. x 8x m n n m 8a 19. b b 0b 0b x 0. 6x 9 18x 7x 1. x x a. a a 9 x. x 1 x x 1 7x. x x 1x 9. y 6 y y y 0 6. z 6 z z z 1 If the given expressions were used as denominators of rational expressions, then what would be the LCD for each group of denominators See Example. 7. 1, , 9. 1, 18, 0 0., 0, a,1a. 18x,0xy. a b,ab 6, a b. m nw, 6mn w 8,9m 6 nw. x 16, x 8x x 9, x 6x 9 7. x, x, x 8. y, y, y 9. x x, x 16, x 0. y, y y, y Find the LCD for the given rational expressions, and convert each rational expression into an equivalent rational expression with the LCD as the denominator. See Example , 8., 1 0., 8 a 6 b b 6., 7 a 10 ab 1., x x 6., 8a b 9 6a c x 1 7.,, 9y z 1 yx 6x y b 8.,, 1 a 6 b 1 a a 1b In Exercises 9 60, find the LCD for the given rational expressions, and convert each rational expression into an equivalent rational expression with the LCD as the denominator. See Example. x x 9., x x a a 0., a a

7 6 (7 ) Chapter 7 Rational Expressions 1., a 6 6 a x., x y y x x x., x 9 x 6x 9 x., x 1 x x 1 9. q,, q q 9q q q 1 p 60.,, p 7p 1 p 11p 1 p p 0 w. w w, w 1 w w z 1 z 1 6., z 6z 8 z z 6 GETTING MORE INVOLVED 61. Discussion. Why do we learn how to convert two rational expressions into equivalent rational expressions with the same denominator x 7.,, 6x 1 x x b 8. b,, 9 b b b 6. Discussion. Which expression is the LCD for x 1 x 7 and x (x ) x(x ) a) x(x ) b) 6x(x ) c) 6x (x ) d) x (x ) 7. ADDITION AND SUBTRACTION In this section Addition and Subtraction of Rational Numbers Addition and Subtraction of Rational Expressions Applications In Section 7. you learned how to find the LCD and build up the denominators of rational expressions. In this section we will use that knowledge to add and subtract rational expressions with different denominators. Addition and Subtraction of Rational Numbers We can add or subtract rational numbers (or fractions) only with identical denominators according to the following definition. Addition and Subtraction of Rational Numbers If b 0, then a b c b a c and a b b c a c. b b

6.3. section. Building Up the Denominator. To convert the fraction 2 3 factor 21 as 21 3 7. Because 2 3

6.3. section. Building Up the Denominator. To convert the fraction 2 3 factor 21 as 21 3 7. Because 2 3 0 (6-18) Chapter 6 Rational Epressions GETTING MORE INVOLVED 7. Discussion. Evaluate each epression. a) One-half of 1 b) One-third of c) One-half of d) One-half of 1 a) b) c) d) 8 7. Eploration. Let R

More information

SIMPLIFYING SQUARE ROOTS

SIMPLIFYING SQUARE ROOTS 40 (8-8) Chapter 8 Powers and Roots 8. SIMPLIFYING SQUARE ROOTS In this section Using the Product Rule Rationalizing the Denominator Simplified Form of a Square Root In Section 8. you learned to simplify

More information

Simplification of Radical Expressions

Simplification of Radical Expressions 8. Simplification of Radical Expressions 8. OBJECTIVES 1. Simplify a radical expression by using the product property. Simplify a radical expression by using the quotient property NOTE A precise set of

More information

FACTORING OUT COMMON FACTORS

FACTORING OUT COMMON FACTORS 278 (6 2) Chapter 6 Factoring 6.1 FACTORING OUT COMMON FACTORS In this section Prime Factorization of Integers Greatest Common Factor Finding the Greatest Common Factor for Monomials Factoring Out the

More information

Negative Integer Exponents

Negative Integer Exponents 7.7 Negative Integer Exponents 7.7 OBJECTIVES. Define the zero exponent 2. Use the definition of a negative exponent to simplify an expression 3. Use the properties of exponents to simplify expressions

More information

Simplifying Algebraic Fractions

Simplifying Algebraic Fractions 5. Simplifying Algebraic Fractions 5. OBJECTIVES. Find the GCF for two monomials and simplify a fraction 2. Find the GCF for two polynomials and simplify a fraction Much of our work with algebraic fractions

More information

FACTORING ax 2 bx c. Factoring Trinomials with Leading Coefficient 1

FACTORING ax 2 bx c. Factoring Trinomials with Leading Coefficient 1 5.7 Factoring ax 2 bx c (5-49) 305 5.7 FACTORING ax 2 bx c In this section In Section 5.5 you learned to factor certain special polynomials. In this section you will learn to factor general quadratic polynomials.

More information

FACTORING POLYNOMIALS

FACTORING POLYNOMIALS 296 (5-40) Chapter 5 Exponents and Polynomials where a 2 is the area of the square base, b 2 is the area of the square top, and H is the distance from the base to the top. Find the volume of a truncated

More information

Factoring (pp. 1 of 4)

Factoring (pp. 1 of 4) Factoring (pp. 1 of 4) Algebra Review Try these items from middle school math. A) What numbers are the factors of 4? B) Write down the prime factorization of 7. C) 6 Simplify 48 using the greatest common

More information

5.1 FACTORING OUT COMMON FACTORS

5.1 FACTORING OUT COMMON FACTORS C H A P T E R 5 Factoring he sport of skydiving was born in the 1930s soon after the military began using parachutes as a means of deploying troops. T Today, skydiving is a popular sport around the world.

More information

Radicals - Rational Exponents

Radicals - Rational Exponents 8. Radicals - Rational Exponents Objective: Convert between radical notation and exponential notation and simplify expressions with rational exponents using the properties of exponents. When we simplify

More information

6.1 The Greatest Common Factor; Factoring by Grouping

6.1 The Greatest Common Factor; Factoring by Grouping 386 CHAPTER 6 Factoring and Applications 6.1 The Greatest Common Factor; Factoring by Grouping OBJECTIVES 1 Find the greatest common factor of a list of terms. 2 Factor out the greatest common factor.

More information

Exponents, Radicals, and Scientific Notation

Exponents, Radicals, and Scientific Notation General Exponent Rules: Exponents, Radicals, and Scientific Notation x m x n = x m+n Example 1: x 5 x = x 5+ = x 7 (x m ) n = x mn Example : (x 5 ) = x 5 = x 10 (x m y n ) p = x mp y np Example : (x) =

More information

Chapter 5. Rational Expressions

Chapter 5. Rational Expressions 5.. Simplify Rational Expressions KYOTE Standards: CR ; CA 7 Chapter 5. Rational Expressions Definition. A rational expression is the quotient P Q of two polynomials P and Q in one or more variables, where

More information

1.3 Polynomials and Factoring

1.3 Polynomials and Factoring 1.3 Polynomials and Factoring Polynomials Constant: a number, such as 5 or 27 Variable: a letter or symbol that represents a value. Term: a constant, variable, or the product or a constant and variable.

More information

Factoring - Greatest Common Factor

Factoring - Greatest Common Factor 6.1 Factoring - Greatest Common Factor Objective: Find the greatest common factor of a polynomial and factor it out of the expression. The opposite of multiplying polynomials together is factoring polynomials.

More information

Algebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.

Algebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions. Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear

More information

When factoring, we look for greatest common factor of each term and reverse the distributive property and take out the GCF.

When factoring, we look for greatest common factor of each term and reverse the distributive property and take out the GCF. Factoring: reversing the distributive property. The distributive property allows us to do the following: When factoring, we look for greatest common factor of each term and reverse the distributive property

More information

6.3 FACTORING ax 2 bx c WITH a 1

6.3 FACTORING ax 2 bx c WITH a 1 290 (6 14) Chapter 6 Factoring e) What is the approximate maximum revenue? f) Use the accompanying graph to estimate the price at which the revenue is zero. y Revenue (thousands of dollars) 300 200 100

More information

Radicals - Multiply and Divide Radicals

Radicals - Multiply and Divide Radicals 8. Radicals - Multiply and Divide Radicals Objective: Multiply and divide radicals using the product and quotient rules of radicals. Multiplying radicals is very simple if the index on all the radicals

More information

The Greatest Common Factor; Factoring by Grouping

The Greatest Common Factor; Factoring by Grouping 296 CHAPTER 5 Factoring and Applications 5.1 The Greatest Common Factor; Factoring by Grouping OBJECTIVES 1 Find the greatest common factor of a list of terms. 2 Factor out the greatest common factor.

More information

Factoring Special Polynomials

Factoring Special Polynomials 6.6 Factoring Special Polynomials 6.6 OBJECTIVES 1. Factor the difference of two squares 2. Factor the sum or difference of two cubes In this section, we will look at several special polynomials. These

More information

Sect 6.1 - Greatest Common Factor and Factoring by Grouping

Sect 6.1 - Greatest Common Factor and Factoring by Grouping Sect 6.1 - Greatest Common Factor and Factoring by Grouping Our goal in this chapter is to solve non-linear equations by breaking them down into a series of linear equations that we can solve. To do this,

More information

Section 1. Finding Common Terms

Section 1. Finding Common Terms Worksheet 2.1 Factors of Algebraic Expressions Section 1 Finding Common Terms In worksheet 1.2 we talked about factors of whole numbers. Remember, if a b = ab then a is a factor of ab and b is a factor

More information

Definition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.

Definition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality. 8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections 4.6 and 1.1) 8.1 Equivalent Inequalities Definition 8.1 Two inequalities are equivalent

More information

Greatest Common Factor (GCF) Factoring

Greatest Common Factor (GCF) Factoring Section 4 4: Greatest Common Factor (GCF) Factoring The last chapter introduced the distributive process. The distributive process takes a product of a monomial and a polynomial and changes the multiplication

More information

MULTIPLICATION AND DIVISION OF REAL NUMBERS In this section we will complete the study of the four basic operations with real numbers.

MULTIPLICATION AND DIVISION OF REAL NUMBERS In this section we will complete the study of the four basic operations with real numbers. 1.4 Multiplication and (1-25) 25 In this section Multiplication of Real Numbers Division by Zero helpful hint The product of two numbers with like signs is positive, but the product of three numbers with

More information

Adding and Subtracting Fractions. 1. The denominator of a fraction names the fraction. It tells you how many equal parts something is divided into.

Adding and Subtracting Fractions. 1. The denominator of a fraction names the fraction. It tells you how many equal parts something is divided into. Tallahassee Community College Adding and Subtracting Fractions Important Ideas:. The denominator of a fraction names the fraction. It tells you how many equal parts something is divided into.. The numerator

More information

MATH 10034 Fundamental Mathematics IV

MATH 10034 Fundamental Mathematics IV MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.

More information

7-2 Factoring by GCF. Warm Up Lesson Presentation Lesson Quiz. Holt McDougal Algebra 1

7-2 Factoring by GCF. Warm Up Lesson Presentation Lesson Quiz. Holt McDougal Algebra 1 7-2 Factoring by GCF Warm Up Lesson Presentation Lesson Quiz Algebra 1 Warm Up Simplify. 1. 2(w + 1) 2. 3x(x 2 4) 2w + 2 3x 3 12x Find the GCF of each pair of monomials. 3. 4h 2 and 6h 2h 4. 13p and 26p

More information

Operations with Algebraic Expressions: Multiplication of Polynomials

Operations with Algebraic Expressions: Multiplication of Polynomials Operations with Algebraic Expressions: Multiplication of Polynomials The product of a monomial x monomial To multiply a monomial times a monomial, multiply the coefficients and add the on powers with the

More information

Factoring - Grouping

Factoring - Grouping 6.2 Factoring - Grouping Objective: Factor polynomials with four terms using grouping. The first thing we will always do when factoring is try to factor out a GCF. This GCF is often a monomial like in

More information

2.3. Finding polynomial functions. An Introduction:

2.3. Finding polynomial functions. An Introduction: 2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned

More information

Prime Factorization, Greatest Common Factor (GCF), and Least Common Multiple (LCM)

Prime Factorization, Greatest Common Factor (GCF), and Least Common Multiple (LCM) Prime Factorization, Greatest Common Factor (GCF), and Least Common Multiple (LCM) Definition of a Prime Number A prime number is a whole number greater than 1 AND can only be divided evenly by 1 and itself.

More information

Tool 1. Greatest Common Factor (GCF)

Tool 1. Greatest Common Factor (GCF) Chapter 4: Factoring Review Tool 1 Greatest Common Factor (GCF) This is a very important tool. You must try to factor out the GCF first in every problem. Some problems do not have a GCF but many do. When

More information

Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.

Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}

More information

Rational Expressions - Least Common Denominators

Rational Expressions - Least Common Denominators 7.3 Rational Expressions - Least Common Denominators Objective: Idenfity the least common denominator and build up denominators to match this common denominator. As with fractions, the least common denominator

More information

A Systematic Approach to Factoring

A Systematic Approach to Factoring A Systematic Approach to Factoring Step 1 Count the number of terms. (Remember****Knowing the number of terms will allow you to eliminate unnecessary tools.) Step 2 Is there a greatest common factor? Tool

More information

3.1. RATIONAL EXPRESSIONS

3.1. RATIONAL EXPRESSIONS 3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers

More information

By reversing the rules for multiplication of binomials from Section 4.6, we get rules for factoring polynomials in certain forms.

By reversing the rules for multiplication of binomials from Section 4.6, we get rules for factoring polynomials in certain forms. SECTION 5.4 Special Factoring Techniques 317 5.4 Special Factoring Techniques OBJECTIVES 1 Factor a difference of squares. 2 Factor a perfect square trinomial. 3 Factor a difference of cubes. 4 Factor

More information

A.2. Exponents and Radicals. Integer Exponents. What you should learn. Exponential Notation. Why you should learn it. Properties of Exponents

A.2. Exponents and Radicals. Integer Exponents. What you should learn. Exponential Notation. Why you should learn it. Properties of Exponents Appendix A. Exponents and Radicals A11 A. Exponents and Radicals What you should learn Use properties of exponents. Use scientific notation to represent real numbers. Use properties of radicals. Simplify

More information

3 cups ¾ ½ ¼ 2 cups ¾ ½ ¼. 1 cup ¾ ½ ¼. 1 cup. 1 cup ¾ ½ ¼ ¾ ½ ¼. 1 cup. 1 cup ¾ ½ ¼ ¾ ½ ¼

3 cups ¾ ½ ¼ 2 cups ¾ ½ ¼. 1 cup ¾ ½ ¼. 1 cup. 1 cup ¾ ½ ¼ ¾ ½ ¼. 1 cup. 1 cup ¾ ½ ¼ ¾ ½ ¼ cups cups cup Fractions are a form of division. When I ask what is / I am asking How big will each part be if I break into equal parts? The answer is. This a fraction. A fraction is part of a whole. The

More information

Rational Exponents. Squaring both sides of the equation yields. and to be consistent, we must have

Rational Exponents. Squaring both sides of the equation yields. and to be consistent, we must have 8.6 Rational Exponents 8.6 OBJECTIVES 1. Define rational exponents 2. Simplify expressions containing rational exponents 3. Use a calculator to estimate the value of an expression containing rational exponents

More information

Factoring a Difference of Two Squares. Factoring a Difference of Two Squares

Factoring a Difference of Two Squares. Factoring a Difference of Two Squares 284 (6 8) Chapter 6 Factoring 87. Tomato soup. The amount of metal S (in square inches) that it takes to make a can for tomato soup is a function of the radius r and height h: S 2 r 2 2 rh a) Rewrite this

More information

Factoring Polynomials

Factoring Polynomials UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can

More information

SIMPLIFYING ALGEBRAIC FRACTIONS

SIMPLIFYING ALGEBRAIC FRACTIONS Tallahassee Community College 5 SIMPLIFYING ALGEBRAIC FRACTIONS In arithmetic, you learned that a fraction is in simplest form if the Greatest Common Factor (GCF) of the numerator and the denominator is

More information

9.3 OPERATIONS WITH RADICALS

9.3 OPERATIONS WITH RADICALS 9. Operations with Radicals (9 1) 87 9. OPERATIONS WITH RADICALS In this section Adding and Subtracting Radicals Multiplying Radicals Conjugates In this section we will use the ideas of Section 9.1 in

More information

1.2 Linear Equations and Rational Equations

1.2 Linear Equations and Rational Equations Linear Equations and Rational Equations Section Notes Page In this section, you will learn how to solve various linear and rational equations A linear equation will have an variable raised to a power of

More information

Chapter 3 Section 6 Lesson Polynomials

Chapter 3 Section 6 Lesson Polynomials Chapter Section 6 Lesson Polynomials Introduction This lesson introduces polynomials and like terms. As we learned earlier, a monomial is a constant, a variable, or the product of constants and variables.

More information

Click on the links below to jump directly to the relevant section

Click on the links below to jump directly to the relevant section Click on the links below to jump directly to the relevant section What is algebra? Operations with algebraic terms Mathematical properties of real numbers Order of operations What is Algebra? Algebra is

More information

0.8 Rational Expressions and Equations

0.8 Rational Expressions and Equations 96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions - that is, algebraic fractions - and equations which contain them. The reader is encouraged to

More information

Solutions of Linear Equations in One Variable

Solutions of Linear Equations in One Variable 2. Solutions of Linear Equations in One Variable 2. OBJECTIVES. Identify a linear equation 2. Combine like terms to solve an equation We begin this chapter by considering one of the most important tools

More information

Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers. Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

More information

Math 10C. Course: Polynomial Products and Factors. Unit of Study: Step 1: Identify the Outcomes to Address. Guiding Questions:

Math 10C. Course: Polynomial Products and Factors. Unit of Study: Step 1: Identify the Outcomes to Address. Guiding Questions: Course: Unit of Study: Math 10C Polynomial Products and Factors Step 1: Identify the Outcomes to Address Guiding Questions: What do I want my students to learn? What can they currently understand and do?

More information

Chapter R.4 Factoring Polynomials

Chapter R.4 Factoring Polynomials Chapter R.4 Factoring Polynomials Introduction to Factoring To factor an expression means to write the expression as a product of two or more factors. Sample Problem: Factor each expression. a. 15 b. x

More information

Factoring Flow Chart

Factoring Flow Chart Factoring Flow Chart greatest common factor? YES NO factor out GCF leaving GCF(quotient) how many terms? 4+ factor by grouping 2 3 difference of squares? perfect square trinomial? YES YES NO NO a 2 -b

More information

In the above, the number 19 is an example of a number because its only positive factors are one and itself.

In the above, the number 19 is an example of a number because its only positive factors are one and itself. Math 100 Greatest Common Factor and Factoring by Grouping (Review) Factoring Definition: A factor is a number, variable, monomial, or polynomial which is multiplied by another number, variable, monomial,

More information

SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS

SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS (Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic

More information

POLYNOMIALS and FACTORING

POLYNOMIALS and FACTORING POLYNOMIALS and FACTORING Exponents ( days); 1. Evaluate exponential expressions. Use the product rule for exponents, 1. How do you remember the rules for exponents?. How do you decide which rule to use

More information

Polynomials. Key Terms. quadratic equation parabola conjugates trinomial. polynomial coefficient degree monomial binomial GCF

Polynomials. Key Terms. quadratic equation parabola conjugates trinomial. polynomial coefficient degree monomial binomial GCF Polynomials 5 5.1 Addition and Subtraction of Polynomials and Polynomial Functions 5.2 Multiplication of Polynomials 5.3 Division of Polynomials Problem Recognition Exercises Operations on Polynomials

More information

7-6. Choosing a Factoring Model. Extension: Factoring Polynomials with More Than One Variable IN T RO DUC E T EACH. Standards for Mathematical Content

7-6. Choosing a Factoring Model. Extension: Factoring Polynomials with More Than One Variable IN T RO DUC E T EACH. Standards for Mathematical Content 7-6 Choosing a Factoring Model Extension: Factoring Polynomials with More Than One Variable Essential question: How can you factor polynomials with more than one variable? What is the connection between

More information

Factoring Methods. Example 1: 2x + 2 2 * x + 2 * 1 2(x + 1)

Factoring Methods. Example 1: 2x + 2 2 * x + 2 * 1 2(x + 1) Factoring Methods When you are trying to factor a polynomial, there are three general steps you want to follow: 1. See if there is a Greatest Common Factor 2. See if you can Factor by Grouping 3. See if

More information

To Evaluate an Algebraic Expression

To Evaluate an Algebraic Expression 1.5 Evaluating Algebraic Expressions 1.5 OBJECTIVES 1. Evaluate algebraic expressions given any signed number value for the variables 2. Use a calculator to evaluate algebraic expressions 3. Find the sum

More information

Chapter 7 - Roots, Radicals, and Complex Numbers

Chapter 7 - Roots, Radicals, and Complex Numbers Math 233 - Spring 2009 Chapter 7 - Roots, Radicals, and Complex Numbers 7.1 Roots and Radicals 7.1.1 Notation and Terminology In the expression x the is called the radical sign. The expression under the

More information

HFCC Math Lab Arithmetic - 4. Addition, Subtraction, Multiplication and Division of Mixed Numbers

HFCC Math Lab Arithmetic - 4. Addition, Subtraction, Multiplication and Division of Mixed Numbers HFCC Math Lab Arithmetic - Addition, Subtraction, Multiplication and Division of Mixed Numbers Part I: Addition and Subtraction of Mixed Numbers There are two ways of adding and subtracting mixed numbers.

More information

COLLEGE ALGEBRA. Paul Dawkins

COLLEGE ALGEBRA. Paul Dawkins COLLEGE ALGEBRA Paul Dawkins Table of Contents Preface... iii Outline... iv Preliminaries... Introduction... Integer Exponents... Rational Exponents... 9 Real Exponents...5 Radicals...6 Polynomials...5

More information

Vocabulary Words and Definitions for Algebra

Vocabulary Words and Definitions for Algebra Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms

More information

FRACTIONS COMMON MISTAKES

FRACTIONS COMMON MISTAKES FRACTIONS COMMON MISTAKES 0/0/009 Fractions Changing Fractions to Decimals How to Change Fractions to Decimals To change fractions to decimals, you need to divide the numerator (top number) by the denominator

More information

Chapter 9. Systems of Linear Equations

Chapter 9. Systems of Linear Equations Chapter 9. Systems of Linear Equations 9.1. Solve Systems of Linear Equations by Graphing KYOTE Standards: CR 21; CA 13 In this section we discuss how to solve systems of two linear equations in two variables

More information

Rules of Exponents. Math at Work: Motorcycle Customization OUTLINE CHAPTER

Rules of Exponents. Math at Work: Motorcycle Customization OUTLINE CHAPTER Rules of Exponents CHAPTER 5 Math at Work: Motorcycle Customization OUTLINE Study Strategies: Taking Math Tests 5. Basic Rules of Exponents Part A: The Product Rule and Power Rules Part B: Combining the

More information

SUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills

SUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E)

More information

Algebra Cheat Sheets

Algebra Cheat Sheets Sheets Algebra Cheat Sheets provide you with a tool for teaching your students note-taking, problem-solving, and organizational skills in the context of algebra lessons. These sheets teach the concepts

More information

Radicals - Rationalize Denominators

Radicals - Rationalize Denominators 8. Radicals - Rationalize Denominators Objective: Rationalize the denominators of radical expressions. It is considered bad practice to have a radical in the denominator of a fraction. When this happens

More information

Name Intro to Algebra 2. Unit 1: Polynomials and Factoring

Name Intro to Algebra 2. Unit 1: Polynomials and Factoring Name Intro to Algebra 2 Unit 1: Polynomials and Factoring Date Page Topic Homework 9/3 2 Polynomial Vocabulary No Homework 9/4 x In Class assignment None 9/5 3 Adding and Subtracting Polynomials Pg. 332

More information

SPECIAL PRODUCTS AND FACTORS

SPECIAL PRODUCTS AND FACTORS CHAPTER 442 11 CHAPTER TABLE OF CONTENTS 11-1 Factors and Factoring 11-2 Common Monomial Factors 11-3 The Square of a Monomial 11-4 Multiplying the Sum and the Difference of Two Terms 11-5 Factoring the

More information

( ) FACTORING. x In this polynomial the only variable in common to all is x.

( ) FACTORING. x In this polynomial the only variable in common to all is x. FACTORING Factoring is similar to breaking up a number into its multiples. For example, 10=5*. The multiples are 5 and. In a polynomial it is the same way, however, the procedure is somewhat more complicated

More information

1.3 Algebraic Expressions

1.3 Algebraic Expressions 1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,

More information

Mathematics Placement

Mathematics Placement Mathematics Placement The ACT COMPASS math test is a self-adaptive test, which potentially tests students within four different levels of math including pre-algebra, algebra, college algebra, and trigonometry.

More information

15.1 Factoring Polynomials

15.1 Factoring Polynomials LESSON 15.1 Factoring Polynomials Use the structure of an expression to identify ways to rewrite it. Also A.SSE.3? ESSENTIAL QUESTION How can you use the greatest common factor to factor polynomials? EXPLORE

More information

Algebra I Vocabulary Cards

Algebra I Vocabulary Cards Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression

More information

This is Factoring and Solving by Factoring, chapter 6 from the book Beginning Algebra (index.html) (v. 1.0).

This is Factoring and Solving by Factoring, chapter 6 from the book Beginning Algebra (index.html) (v. 1.0). This is Factoring and Solving by Factoring, chapter 6 from the book Beginning Algebra (index.html) (v. 1.0). This book is licensed under a Creative Commons by-nc-sa 3.0 (http://creativecommons.org/licenses/by-nc-sa/

More information

Five 5. Rational Expressions and Equations C H A P T E R

Five 5. Rational Expressions and Equations C H A P T E R Five C H A P T E R Rational Epressions and Equations. Rational Epressions and Functions. Multiplication and Division of Rational Epressions. Addition and Subtraction of Rational Epressions.4 Comple Fractions.

More information

Solving Rational Equations

Solving Rational Equations Lesson M Lesson : Student Outcomes Students solve rational equations, monitoring for the creation of extraneous solutions. Lesson Notes In the preceding lessons, students learned to add, subtract, multiply,

More information

Algebra 2 PreAP. Name Period

Algebra 2 PreAP. Name Period Algebra 2 PreAP Name Period IMPORTANT INSTRUCTIONS FOR STUDENTS!!! We understand that students come to Algebra II with different strengths and needs. For this reason, students have options for completing

More information

Activity 1: Using base ten blocks to model operations on decimals

Activity 1: Using base ten blocks to model operations on decimals Rational Numbers 9: Decimal Form of Rational Numbers Objectives To use base ten blocks to model operations on decimal numbers To review the algorithms for addition, subtraction, multiplication and division

More information

BEGINNING ALGEBRA ACKNOWLEDMENTS

BEGINNING ALGEBRA ACKNOWLEDMENTS BEGINNING ALGEBRA The Nursing Department of Labouré College requested the Department of Academic Planning and Support Services to help with mathematics preparatory materials for its Bachelor of Science

More information

This is a square root. The number under the radical is 9. (An asterisk * means multiply.)

This is a square root. The number under the radical is 9. (An asterisk * means multiply.) Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize

More information

expression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method.

expression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method. A polynomial of degree n (in one variable, with real coefficients) is an expression of the form: a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 2 x 2 + a 1 x + a 0 where a n, a n 1, a n 2, a 2, a 1, a 0 are

More information

Unit 7: Radical Functions & Rational Exponents

Unit 7: Radical Functions & Rational Exponents Date Period Unit 7: Radical Functions & Rational Exponents DAY 0 TOPIC Roots and Radical Expressions Multiplying and Dividing Radical Expressions Binomial Radical Expressions Rational Exponents 4 Solving

More information

Method To Solve Linear, Polynomial, or Absolute Value Inequalities:

Method To Solve Linear, Polynomial, or Absolute Value Inequalities: Solving Inequalities An inequality is the result of replacing the = sign in an equation with ,, or. For example, 3x 2 < 7 is a linear inequality. We call it linear because if the < were replaced with

More information

Simplification Problems to Prepare for Calculus

Simplification Problems to Prepare for Calculus Simplification Problems to Prepare for Calculus In calculus, you will encounter some long epressions that will require strong factoring skills. This section is designed to help you develop those skills.

More information

3.3 Addition and Subtraction of Rational Numbers

3.3 Addition and Subtraction of Rational Numbers 3.3 Addition and Subtraction of Rational Numbers In this section we consider addition and subtraction of both fractions and decimals. We start with addition and subtraction of fractions with the same denominator.

More information

Radicals - Multiply and Divide Radicals

Radicals - Multiply and Divide Radicals 8. Radicals - Multiply and Divide Radicals Objective: Multiply and divide radicals using the product and quotient rules of radicals. Multiplying radicals is very simple if the index on all the radicals

More information

2.3 Solving Equations Containing Fractions and Decimals

2.3 Solving Equations Containing Fractions and Decimals 2. Solving Equations Containing Fractions and Decimals Objectives In this section, you will learn to: To successfully complete this section, you need to understand: Solve equations containing fractions

More information

Exponents. Learning Objectives 4-1

Exponents. Learning Objectives 4-1 Eponents -1 to - Learning Objectives -1 The product rule for eponents The quotient rule for eponents The power rule for eponents Power rules for products and quotient We can simplify by combining the like

More information

COWLEY COUNTY COMMUNITY COLLEGE REVIEW GUIDE Compass Algebra Level 2

COWLEY COUNTY COMMUNITY COLLEGE REVIEW GUIDE Compass Algebra Level 2 COWLEY COUNTY COMMUNITY COLLEGE REVIEW GUIDE Compass Algebra Level This study guide is for students trying to test into College Algebra. There are three levels of math study guides. 1. If x and y 1, what

More information

Clifton High School Mathematics Summer Workbook Algebra 1

Clifton High School Mathematics Summer Workbook Algebra 1 1 Clifton High School Mathematics Summer Workbook Algebra 1 Completion of this summer work is required on the first day of the school year. Date Received: Date Completed: Student Signature: Parent Signature:

More information

FRACTION WORKSHOP. Example: Equivalent Fractions fractions that have the same numerical value even if they appear to be different.

FRACTION WORKSHOP. Example: Equivalent Fractions fractions that have the same numerical value even if they appear to be different. FRACTION WORKSHOP Parts of a Fraction: Numerator the top of the fraction. Denominator the bottom of the fraction. In the fraction the numerator is 3 and the denominator is 8. Equivalent Fractions: Equivalent

More information

MATH 21. College Algebra 1 Lecture Notes

MATH 21. College Algebra 1 Lecture Notes MATH 21 College Algebra 1 Lecture Notes MATH 21 3.6 Factoring Review College Algebra 1 Factoring and Foiling 1. (a + b) 2 = a 2 + 2ab + b 2. 2. (a b) 2 = a 2 2ab + b 2. 3. (a + b)(a b) = a 2 b 2. 4. (a

More information

Transition To College Mathematics

Transition To College Mathematics Transition To College Mathematics In Support of Kentucky s College and Career Readiness Program Northern Kentucky University Kentucky Online Testing (KYOTE) Group Steve Newman Mike Waters Janis Broering

More information