Section A3 Polynomials: Factoring APPLICATIONS. A22 Appendix A A BASIC ALGEBRA REVIEW


 Oliver Matthew Anderson
 5 years ago
 Views:
Transcription
1 A Appendi A A BASIC ALGEBRA REVIEW C In Problems 53 56, perform the indicated operations and simplify. 53. ( ) 3 ( ) 3( ) ( ) 3 ( ) 3( ) {[ ( )] ( )( 3)} 56. {( 3)( ) [3 ( )]} 57. Show by eample that, in general, (a b) a b. Discuss possible conditions on a and b that would make this a valid equation. 58. Show by eample that, in general, (a b) a b. Discuss possible conditions on a and b that would make this a valid equation. 59. If you are given two polynomials, one of degree m and the other of degree n, m n, what is the degree of the sum? 60. What is the degree of the product of the two polynomials in Problem 59? 6. How does the answer to Problem 59 change if the two polynomials can have the same degree? 6. How does the answer to Problem 60 change if the two polynomials can have the same degree? 65. Coin Problem. A parking meter contains nickels, dimes, and quarters. There are 5 fewer dimes than nickels, and more quarters than dimes. If represents the number of nickels, write an algebraic epression in terms of that represents the value of all the coins in the meter in cents. Simplify the epression. 66. Coin Problem. A vending machine contains dimes and quarters only. There are 4 more dimes than quarters. If represents the number of quarters, write an algebraic epression in terms of that represents the value of all the coins in the vending machine in cents. Simplify the epression. 67. Packaging. A spherical plastic container for designer wristwatches has an inner radius of centimeters (see the figure). If the plastic shell is 0.3 centimeters thick, write an algebraic epression in terms of that represents the volume of the plastic used to construct the container. Simplify the epression. [Recall: The volume V of a sphere of 4 radius r is given by V r 3.] cm cm APPLICATIONS 63. Geometry. The width of a rectangle is 5 centimeters less than its length. If represents the length, write an algebraic epression in terms of that represents the perimeter of the rectangle. Simplify the epression. 64. Geometry. The length of a rectangle is 8 meters more than its width. If represents the width of the rectangle, write an algebraic epression in terms of that represents its area. Change the epression to a form without parentheses. 68. Packaging. A cubical container for shipping computer components is formed by coating a metal mold with polystyrene. If the metal mold is a cube with sides centimeters long and the polystyrene coating is centimeters thick, write an algebraic epression in terms of that represents the volume of the polystyrene used to construct the container. Simplify the epression. [Recall: The volume V of a cube with sides of length t is given by V t 3.] Section A3 Polynomials: Factoring Factoring What Does It Mean? Common Factors and Factoring by Grouping Factoring SecondDegree Polynomials More Factoring Factoring_What Does It Mean? A factor of a number is one of two or more numbers whose product is the given number. Similarly, a factor of an algebraic epression is one of two or more algebraic epressions whose product is the given algebraic epression. For eample,
2 A3 Polynomials: Factoring A , 3, and 5 are each factors of ( )( ) ( ) and ( ) are each factors of 4. The process of writing a number or algebraic epression as the product of other numbers or algebraic epressions is called factoring. We start our discussion of factoring with the positive integers. An integer such as 30 can be represented in a factored form in many ways. The products 6 5 ( )(0)(6) all yield 30. A particularly useful way of factoring positive integers greater than is in terms of prime numbers. DEFINITION PRIME AND COMPOSITE NUMBERS An integer greater than is prime if its only positive integer factors are itself and. An integer greater than that is not prime is called a composite number. The integer is neither prime nor composite. Eamples of prime numbers:, 3, 5, 7,, 3 Eamples of composite numbers: 4, 6, 8, 9, 0, Eplore/Discuss In the array below, cross out all multiples of, ecept itself. Then cross out all multiples of 3, ecept 3 itself. Repeat this for each integer in the array that has not yet been crossed out. Describe the set of numbers that remains when this process is completed This process is referred to as the sieve of Eratosthenes. (Eratosthenes was a Greek mathematician and astronomer who was a contemporary of Archimedes, circa 00 B.C.) A composite number is said to be factored completely if it is represented as a product of prime factors. The only factoring of 30 given above that meets this condition is
3 A4 Appendi A A BASIC ALGEBRA REVIEW Factoring a Composite Number Write 60 in completely factored form. Solution or or Write 80 in completely factored form. Notice in Eample that we end up with the same prime factors for 60 irrespective of how we progress through the factoring process. This illustrates an important property of integers: THEOREM THE FUNDAMENTAL THEOREM OF ARITHMETIC Each integer greater than is either prime or can be epressed uniquely, ecept for the order of factors, as a product of prime factors. We can also write polynomials in completely factored form. A polynomial such as 6 can be written in factored form in many ways. The products ( 3)( ) ( 3) ( )( ) all yield 6. A particularly useful way of factoring polynomials is in terms of prime polynomials. 3 DEFINITION PRIME POLYNOMIALS A polynomial of degree greater than 0 is said to be prime relative to a given set of numbers if: () all of its coefficients are from that set of numbers; and () it cannot be written as a product of two polynomials, ecluding and itself, having coefficients from that set of numbers. Relative to the set of integers: is prime 9 is not prime, since 9 ( 3)( 3) [Note: The set of numbers most frequently used in factoring polynomials is the set of integers.]
4 A3 Polynomials: Factoring A5 A nonprime polynomial is said to be factored completely relative to a given set of numbers if it is written as a product of prime polynomials relative to that set of numbers. Our objective in this section is to review some of the standard factoring techniques for polynomials with integer coefficients. In Chapter 3 we treated in detail the topic of factoring polynomials of higher degree with arbitrary coefficients. Common Factors and Factoring by Grouping The net eample illustrates the use of the distributive properties in factoring. Factoring Out Common Factors Factor out, relative to the integers, all factors common to all terms. (A) 3 y 8 y 6y 3 (B) (3 ) 7(3 ) Solutions (A) 3 y 8 y 6y 3 (y) (y)4y (y)3y y( 4y 3y ) (B) (3 ) 7(3 ) (3 ) 7(3 ) ( 7)(3 ) Factor out, relative to the integers, all factors common to all terms. (A) 3 3 y 6 y 3y 3 (B) 3y(y 5) (y 5) 3 Factoring Out Common Factors Factor completely relative to the integers: 4( 7)( 3) ( 7) ( 3) Solution 4( 7)( 3) ( 7) ( 3) ( 7)( 3)[( 3) ( 7)] ( 7)( 3)( 6 7) ( 7)( 3)(4 ) 3 Factor completely relative to the integers. 4( 5)(3 ) 6( 5) (3 ) Some polynomials can be factored by first grouping terms in such a way that we obtain an algebraic epression that looks something like Eample, part B. We can then complete the factoring by the method used in that eample.
5 A6 Appendi A A BASIC ALGEBRA REVIEW 4 Factoring by Grouping Factor completely, relative to the integers, by grouping. (A) (B) wy wz y z (C) 3ac bd 3ad bc Solutions (A) (3 6) (4 8) 3( ) 4( ) (3 4)( ) (B) wy wz y z (wy wz) (y z) w( y z) ( y z) (w )( y z) Group the first two and last two terms. Remove common factors from each group. Factor out the common factor ( ). Group the first two and last two terms be careful of signs. Remove common factors from each group. Factor out the common factor (y z). (C) 3ac bd 3ad bc In parts A and B the polynomials are arranged in such a way that grouping the first two terms and the last two terms leads to common factors. In this problem neither the first two terms nor the last two terms have a common factor. Sometimes rearranging terms will lead to a factoring by grouping. In this case, we interchange the second and fourth terms to obtain a problem comparable to part B, which can be factored as follows: 3ac bc 3ad bd (3ac bc) (3ad bd) c(3a b) d(3a b) (c d)(3a b) 4 Factor completely, relative to the integers, by grouping. (A) (B) pr ps 6qr 3qs (C) 6wy z y 3wz Factoring SecondDegree Polynomials We now turn our attention to factoring seconddegree polynomials of the form 5 3 and 3y y into the product of two firstdegree polynomials with integer coefficients. The following eample will illustrate an approach to the problem.
6 A3 Polynomials: Factoring A7 5 Factoring SecondDegree Polynomials Factor each polynomial, if possible, using integer coefficients. (A) 3y y (B) 3 4 (C) 6 5y 4y Solutions (A) 3y y ( y)( y)?? Put in what we know. Signs must be opposite. (We can reverse this choice if we get 3y instead of 3y for the middle term.) Now, what are the factors of (the coefficient of y )? ( y)( y) 3y y ( y)( y) y The first choice gives us 3y for the middle term close, but not there so we reverse our choice of signs to obtain 3y y ( y)( y) (B) 3 4 ( )( ) Signs must be the same because the third term is positive and must be negative because the middle term is negative ( )( ) 4 4 ( )( 4) 5 4 ( 4)( ) 5 4 No choice produces the middle term; hence 3 4 is not factorable using integer coefficients. (C) 6 5y 4y ( y)( y)???? The signs must be opposite in the factors, because the third term is negative. We can reverse our choice of signs later if necessary. We now write all factors of 6 and of 4:
7 A8 Appendi A A BASIC ALGEBRA REVIEW and try each choice on the left with each on the right a total of combinations that give us the first and last terms in the polynomial 6 5y 4y. The question is: Does any combination also give us the middle term, 5y? After trial and error and, perhaps, some educated guessing among the choices, we find that 3 matched with 4 gives us the correct middle term. Thus, 6 5y 4y (3 4y)( y) If none of the 4 combinations (including reversing our sign choice) had produced the middle term, then we would conclude that the polynomial is not factorable using integer coefficients. 5 Factor each polynomial, if possible, using integer coefficients. (A) 8 (B) 5 (C) 7y 4y (D) 4 5y 4y More Factoring The factoring formulas listed below will enable us to factor certain polynomial forms that occur frequently. SPECIAL FACTORING FORMULAS. u uv v (u v) Perfect Square. u uv v (u v) Perfect Square 3. u v (u v)(u v) Difference of Squares 4. u 3 v 3 (u v)(u uv v ) Difference of Cubes 5. u 3 v 3 (u v)(u uv v ) Sum of Cubes The formulas in the bo can be established by multiplying the factors on the right. CAUTION Note that we did not list a special factoring formula for the sum of two squares. In general, u v (au bv)(cu dv) for any choice of real number coefficients a, b, c, and d.
8 A3 Polynomials: Factoring A9 6 Using Special Factoring Formulas Factor completely relative to the integers. (A) 6y 9y (B) 9 4y (C) 8m 3 (D) 3 y 3 z 3 Solutions (A) 6y 9y ()(3y) (3y) ( 3y) (B) 9 4y (3) (y) (3 y)(3 y) (C) 8m 3 (m) 3 3 (m )[(m) (m)() ] (m )(4m m ) (D) 3 y 3 z 3 3 (yz) 3 ( yz)( yz y z ) 6 Factor completely relative to the integers. (A) 4m mn 9n (B) 6y (C) z 3 (D) m 3 n 3 Eplore/Discuss (A) Verify the following factor formulas for u 4 v 4 : u 4 v 4 (u v)(u v)(u v ) (u v)(u 3 u v uv v 3 ) (B) Discuss the pattern in the following formulas: u v (u v)(u v) u 3 v 3 (u v)(u uv v ) u 4 v 4 (u v)(u 3 u v uv v 3 ) (C) Use the pattern you discovered in part B to write similar formulas for u 5 v 5 and u 6 v 6. Verify your formulas by multiplication. We complete this section by considering factoring that involves combinations of the preceding techniques as well as a few additional ones. Generally speaking, When asked to factor a polynomial, we first take out all factors common to all terms, if they are present, and then proceed as above until all factors are prime.
9 A30 Appendi A A BASIC ALGEBRA REVIEW 7 Combining Factoring Techniques Factor completely relative to the integers. (A) (B) 6 9 y (C) 4m 3 n m n mn 3 (D) t 4 6t (E) y 4 5y Solutions (A) (9 4) (3 )(3 ) (B) 6 9 y ( 6 9) y Group the first three terms. ( 3) y Factor 6 9. [( 3) y][( 3) y] ( 3 y)( 3 y) Difference of squares (C) 4m 3 n m n mn 3 mn(m mn n ) (D) t 4 6t t(t 3 8) t(t )(t t 4) (E) y 4 5y (y 3)( y 4) (y 3)( y )( y ) 7 Factor completely relative to the integers. (A) (B) y 4y 4 (C) 3u 4 3u 3 v 9u v (D) 3m 4 4mn 3 (E) Answers to Matched Problems (A) 3y( y y ) (B) (3y )(y 5) 3. ( 5)(3 )( 7) 4. (A) ( 5)( 3) (B) ( p 3q)(r s) (C) (3w )(y z) 5. (A) ( )( 6) (B) Not factorable using integers (C) ( y)( 4y) (D) (4 y)( 4y) 6. (A) (m 3n) (B) ( 4y)( 4y) (C) (z )(z z ) (D) (m n)(m mn n ) 7. (A) 3( 4)( 4) (B) ( y )( y ) (C) 3u (u uv 3v ) (D) 3m(m n)(m mn 4n ) (E) (3 )( )( ) EXERCISE A3 A In Problems 8, factor out, relative to the integers, all factors common to all terms m 4 9m 3 3m y 0 y 5y u 3 v 6u v 4uv ( ) 3( ) 6. 7m(m 3) 5(m 3) 7. w( y z) ( y z) 8. a(3c d ) 4b(3c d ) In Problems 9 6, factor completely relative to integers y 6y 5y 5. 6m 0m 3m y 3y 6y 4. 3a ab ab 8b
10 A3 Polynomials: Factoring A ac 3bd 6bc 4ad 6. 3pr qs qr 6ps In Problems 7 8, factor completely relative to the integers. If a polynomial is prime relative to the integers, say so y y 9. 4y y 0. u uv 5v. 4. m 6m m 6n 4. w y 5. 0y 5y 6. 9m 6mn n 7. u 8 8. y 6 B In Problems 9 44, factor completely relative to the integers. If a polynomial is prime relative to the integers, say so z 8z y 3 y 48y y 8y y 34. 4y y s 7st 3t 36. 6m mn n y 9y u 3 v uv m 3 6m 5m m 3 n 3 4. r 3 t c a 3 Problems are calculusrelated. Factor completely relative to the integers (3 5)( 3) 4(3 5) ( 3) 46. ( 3)(4 7) 8( 3) (4 7) (9 ) (9 ) ( 7) 4 3 ( 7) ( )( 5) 4( ) ( 5) 50. 4( 3) 3 ( ) 3 6( 3) 4 ( ) C In Problems 57 7, factor completely relative to the integers. In polynomials involving more than three terms, try grouping the terms in various combinations as a first step. If a polynomial is prime relative to the integers, say so a 3 a a 60. t 3 t t 6. 4(A B) 5(A B) ( y) 3( y) m 4 n y 4 3y s 4 t 4 8st 66. 7a a 5 b m mn n m n 68. y y y 69. 8a 3 8a( 8 6) 70. 5(4 y 9y ) 9a b a 4 a b b 4 a b APPLICATIONS 73. Construction. A rectangular opentopped bo is to be constructed out of 0inchsquare sheets of thin cardboard by cutting inch squares out of each corner and bending the sides up as indicated in the figure. Epress each of the following quantities as a polynomial in both factored and epanded form. (A) The area of cardboard after the corners have been removed. (B) The volume of the bo. 0 inches 0 inches In Problems 5 56, factor completely relative to the integers. In polynomials involving more than three terms, try grouping the terms in various combinations as a first step. If a polynomial is prime relative to the integers, say so. 5. (a b) 4(c d ) 5. ( ) 9y 53. am 3an bm 3bn 54. 5ac 0ad 3bc 4bd y 4y 56. 5u 4uv v
11 A3 Appendi A A BASIC ALGEBRA REVIEW 74. Construction. A rectangular opentopped bo is to be constructed out of 9 by 6inch sheets of thin cardboard by cutting inch squares out of each corner and bending the sides up. Epress each of the following quantities as a polynomial in both factored and epanded form. (A) The area of cardboard after the corners have been removed. (B) The volume of the bo. Section A4 Rational Epressions: Basic Operations Reducing to Lowest Terms Multiplication and Division Addition and Subtraction Compound Fractions We now turn our attention to fractional forms. A quotient of two algebraic epressions, division by 0 ecluded, is called a fractional epression. If both the numerator and denominator of a fractional epression are polynomials, the fractional epression is called a rational epression. Some eamples of rational epressions are the following (recall, a nonzero constant is a polynomial of degree 0): In this section we discuss basic operations on rational epressions, including multiplication, division, addition, and subtraction. Since variables represent real numbers in the rational epressions we are going to consider, the properties of real number fractions summarized in Section A play a central role in much of the work that we will do. Even though not always eplicitly stated, we always assume that variables are restricted so that division by 0 is ecluded. Reducing to Lowest Terms We start this discussion by restating the fundamental property of fractions (from Theorem 3 in Section A): FUNDAMENTAL PROPERTY OF FRACTIONS If a, b, and k are real numbers with b, k 0, then ka kb a b ( 3) ( 3) 0, 3 Using this property from left to right to eliminate all common factors from the numerator and the denominator of a given fraction is referred to as reducing
SECTION A3 Polynomials: Factoring
A3 Polynomials: Factoring A23 thick, write an algebraic epression in terms of that represents the volume of the plastic used to construct the container. Simplify the epression. [Recall: The volume 4
More informationA.3. Polynomials and Factoring. Polynomials. What you should learn. Definition of a Polynomial in x. Why you should learn it
Appendi A.3 Polynomials and Factoring A23 A.3 Polynomials and Factoring What you should learn Write polynomials in standard form. Add,subtract,and multiply polynomials. Use special products to multiply
More informationFactoring 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 informationPolynomials and Factoring
7.6 Polynomials and Factoring Basic Terminology A term, or monomial, is defined to be a number, a variable, or a product of numbers and variables. A polynomial is a term or a finite sum or difference of
More informationFACTORING 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 informationFACTORING POLYNOMIALS
296 (540) 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 informationNegative 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 informationSECTION P.5 Factoring Polynomials
BLITMCPB.QXP.0599_4874 /0/0 0:4 AM Page 48 48 Chapter P Prerequisites: Fundamental Concepts of Algebra Technology Eercises Critical Thinking Eercises 98. The common cold is caused by a rhinovirus. The
More informationSUNY 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 informationSimplifying 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 informationSPECIAL PRODUCTS AND FACTORS
CHAPTER 442 11 CHAPTER TABLE OF CONTENTS 111 Factors and Factoring 112 Common Monomial Factors 113 The Square of a Monomial 114 Multiplying the Sum and the Difference of Two Terms 115 Factoring the
More informationMultiplying and Dividing Algebraic Fractions
. Multiplying and Dividing Algebraic Fractions. OBJECTIVES. Write the product of two algebraic fractions in simplest form. Write the quotient of two algebraic fractions in simplest form. Simplify a comple
More informationTo 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 informationPOLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a
More informationThis 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 byncsa 3.0 (http://creativecommons.org/licenses/byncsa/
More informationFive 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 informationTopic: Special Products and Factors Subtopic: Rules on finding factors of polynomials
Quarter I: Special Products and Factors and Quadratic Equations Topic: Special Products and Factors Subtopic: Rules on finding factors of polynomials Time Frame: 20 days Time Frame: 3 days Content Standard:
More informationPolynomials. 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 information5.1 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 information3. Solve the equation containing only one variable for that variable.
Question : How do you solve a system of linear equations? There are two basic strategies for solving a system of two linear equations and two variables. In each strategy, one of the variables is eliminated
More informationExponents. Learning Objectives 41
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 informationThe majority of college students hold credit cards. According to the Nellie May
CHAPTER 6 Factoring Polynomials 6.1 The Greatest Common Factor and Factoring by Grouping 6. Factoring Trinomials of the Form b c 6.3 Factoring Trinomials of the Form a b c and Perfect Square Trinomials
More informationMath Review. for the Quantitative Reasoning Measure of the GRE revised General Test
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important
More informationBy 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 informationReview of Intermediate Algebra Content
Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6
More informationSection 5.0A Factoring Part 1
Section 5.0A Factoring Part 1 I. Work Together A. Multiply the following binomials into trinomials. (Write the final result in descending order, i.e., a + b + c ). ( 7)( + 5) ( + 7)( + ) ( + 7)( + 5) (
More informationZero and Negative Exponents and Scientific Notation. a a n a m n. Now, suppose that we allow m to equal n. We then have. a am m a 0 (1) a m
0. E a m p l e 666SECTION 0. OBJECTIVES. Define the zero eponent. Simplif epressions with negative eponents. Write a number in scientific notation. Solve an application of scientific notation We must have
More informationThe Properties of Signed Numbers Section 1.2 The Commutative Properties If a and b are any numbers,
1 Summary DEFINITION/PROCEDURE EXAMPLE REFERENCE From Arithmetic to Algebra Section 1.1 Addition x y means the sum of x and y or x plus y. Some other words The sum of x and 5 is x 5. indicating addition
More information7.2 Quadratic Equations
476 CHAPTER 7 Graphs, Equations, and Inequalities 7. Quadratic Equations Now Work the Are You Prepared? problems on page 48. OBJECTIVES 1 Solve Quadratic Equations by Factoring (p. 476) Solve Quadratic
More informationEquations Involving Fractions
. Equations Involving Fractions. OBJECTIVES. Determine the ecluded values for the variables of an algebraic fraction. Solve a fractional equation. Solve a proportion for an unknown NOTE The resulting equation
More informationCopy 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 informationSECTION 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 information6706_PM10SB_C4_CO_pp192193.qxd 5/8/09 9:53 AM Page 192 192 NEL
92 NEL Chapter 4 Factoring Algebraic Epressions GOALS You will be able to Determine the greatest common factor in an algebraic epression and use it to write the epression as a product Recognize different
More informationSolutions 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 informationSummer Math Exercises. For students who are entering. PreCalculus
Summer Math Eercises For students who are entering PreCalculus It has been discovered that idle students lose learning over the summer months. To help you succeed net fall and perhaps to help you learn
More informationHow To Solve Factoring Problems
05W4801AM1.qxd 8/19/08 8:45 PM Page 241 Factoring, Solving Equations, and Problem Solving 5 5.1 Factoring by Using the Distributive Property 5.2 Factoring the Difference of Two Squares 5.3 Factoring
More informationMATH 102 College Algebra
FACTORING Factoring polnomials ls is simpl the reverse process of the special product formulas. Thus, the reverse process of special product formulas will be used to factor polnomials. To factor polnomials
More informationHow To Factor By Gcf In Algebra 1.5
72 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 informationSimplification 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 information1.3 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 informationPolynomial Degree and Finite Differences
CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial
More information6.1 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 informationMath 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 informationSimplifying Exponential Expressions
Simplifying Eponential Epressions Eponential Notation Base Eponent Base raised to an eponent Eample: What is the base and eponent of the following epression? 7 is the base 7 is the eponent Goal To write
More informationAlum Rock Elementary Union School District Algebra I Study Guide for Benchmark III
Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Name Date Adding and Subtracting Polynomials Algebra Standard 10.0 A polynomial is a sum of one ore more monomials. Polynomial
More informationMATH 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 informationCM2202: Scientific Computing and Multimedia Applications General Maths: 2. Algebra  Factorisation
CM2202: Scientific Computing and Multimedia Applications General Maths: 2. Algebra  Factorisation Prof. David Marshall School of Computer Science & Informatics Factorisation Factorisation is a way of
More information10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED
CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations
More informationFree PreAlgebra Lesson 55! page 1
Free PreAlgebra Lesson 55! page 1 Lesson 55 Perimeter Problems with Related Variables Take your skill at word problems to a new level in this section. All the problems are the same type, so that you can
More information6.3. section. Building Up the Denominator. To convert the fraction 2 3 factor 21 as 21 3 7. Because 2 3
0 (618) Chapter 6 Rational Epressions GETTING MORE INVOLVED 7. Discussion. Evaluate each epression. a) Onehalf of 1 b) Onethird of c) Onehalf of d) Onehalf of 1 a) b) c) d) 8 7. Eploration. Let R
More informationSect. 1.3: Factoring
Sect. 1.3: Factoring MAT 109, Fall 2015 Tuesday, 1 September 2015 Algebraic epression review Epanding algebraic epressions Distributive property a(b + c) = a b + a c (b + c) a = b a + c a Special epansion
More informationVeterans Upward Bound Algebra I Concepts  Honors
Veterans Upward Bound Algebra I Concepts  Honors Brenda Meery Kaitlyn Spong Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) www.ck12.org Chapter 6. Factoring CHAPTER
More informationPolynomial and Synthetic Division. Long Division of Polynomials. Example 1. 6x 2 7x 2 x 2) 19x 2 16x 4 6x3 12x 2 7x 2 16x 7x 2 14x. 2x 4.
_.qd /7/5 9: AM Page 5 Section.. Polynomial and Synthetic Division 5 Polynomial and Synthetic Division What you should learn Use long division to divide polynomials by other polynomials. Use synthetic
More informationHow do you compare numbers? On a number line, larger numbers are to the right and smaller numbers are to the left.
The verbal answers to all of the following questions should be memorized before completion of prealgebra. Answers that are not memorized will hinder your ability to succeed in algebra 1. Number Basics
More informationQuick Reference ebook
This file is distributed FREE OF CHARGE by the publisher Quick Reference Handbooks and the author. Quick Reference ebook Click on Contents or Index in the left panel to locate a topic. The math facts listed
More informationPreCalculus II Factoring and Operations on Polynomials
Factoring... 1 Polynomials...1 Addition of Polynomials... 1 Subtraction of Polynomials...1 Multiplication of Polynomials... Multiplying a monomial by a monomial... Multiplying a monomial by a polynomial...
More informationDecember 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS
December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in twodimensional space (1) 2x y = 3 describes a line in twodimensional space The coefficients of x and y in the equation
More informationA Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions
A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25
More informationA Quick Algebra Review
1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals
More informationMTH 092 College Algebra Essex County College Division of Mathematics Sample Review Questions 1 Created January 17, 2006
MTH 092 College Algebra Essex County College Division of Mathematics Sample Review Questions Created January 7, 2006 Math 092, Elementary Algebra, covers the mathematical content listed below. In order
More informationAnswers to Basic Algebra Review
Answers to Basic Algebra Review 1. 1.1 Follow the sign rules when adding and subtracting: If the numbers have the same sign, add them together and keep the sign. If the numbers have different signs, subtract
More informationFactoring Polynomials
Factoring Polynomials Factoring Factoring is the process of writing a polynomial as the product of two or more polynomials. The factors of 6x 2 x 2 are 2x + 1 and 3x 2. In this section, we will be factoring
More informationNegative Integral Exponents. If x is nonzero, the reciprocal of x is written as 1 x. For example, the reciprocal of 23 is written as 2
4 (4) Chapter 4 Polynomials and Eponents P( r) 0 ( r) dollars. Which law of eponents can be used to simplify the last epression? Simplify it. P( r) 7. CD rollover. Ronnie invested P dollars in a year
More informationRadicals  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 informationFactoring 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 informationBEGINNING 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 informationEAP/GWL Rev. 1/2011 Page 1 of 5. Factoring a polynomial is the process of writing it as the product of two or more polynomial factors.
EAP/GWL Rev. 1/2011 Page 1 of 5 Factoring a polynomial is the process of writing it as the product of two or more polynomial factors. Example: Set the factors of a polynomial equation (as opposed to an
More information1.3 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 informationOperations 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 informationFactoring and Applications
Factoring and Applications What is a factor? The Greatest Common Factor (GCF) To factor a number means to write it as a product (multiplication). Therefore, in the problem 48 3, 4 and 8 are called the
More informationFACTORISATION YEARS. A guide for teachers  Years 9 10 June 2011. The Improving Mathematics Education in Schools (TIMES) Project
9 10 YEARS The Improving Mathematics Education in Schools (TIMES) Project FACTORISATION NUMBER AND ALGEBRA Module 33 A guide for teachers  Years 9 10 June 2011 Factorisation (Number and Algebra : Module
More informationMATH 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 informationAlgebraic 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 information1.4. Arithmetic of Algebraic Fractions. Introduction. Prerequisites. Learning Outcomes
Arithmetic of Algebraic Fractions 1.4 Introduction Just as one whole number divided by another is called a numerical fraction, so one algebraic expression divided by another is known as an algebraic fraction.
More informationCOLLEGE 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 information0.8 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 informationTSI College Level Math Practice Test
TSI College Level Math Practice Test Tutorial Services Mission del Paso Campus. Factor the Following Polynomials 4 a. 6 8 b. c. 7 d. ab + a + b + 6 e. 9 f. 6 9. Perform the indicated operation a. ( +7y)
More information9.3 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 informationLesson 9.1 Solving Quadratic Equations
Lesson 9.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with a. One intercept and all nonnegative yvalues. b. The verte in the third quadrant and no intercepts. c. The verte
More informationExponential and Logarithmic Functions
Chapter 6 Eponential and Logarithmic Functions Section summaries Section 6.1 Composite Functions Some functions are constructed in several steps, where each of the individual steps is a function. For eample,
More informationFactoring  Factoring Special Products
6.5 Factoring  Factoring Special Products Objective: Identify and factor special products including a difference of squares, perfect squares, and sum and difference of cubes. When factoring there are
More informationChapter 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 informationBig Bend Community College. Beginning Algebra MPC 095. Lab Notebook
Big Bend Community College Beginning Algebra MPC 095 Lab Notebook Beginning Algebra Lab Notebook by Tyler Wallace is licensed under a Creative Commons Attribution 3.0 Unported License. Permissions beyond
More informationSOLVING EQUATIONS WITH RADICALS AND EXPONENTS 9.5. section ( 3 5 3 2 )( 3 25 3 10 3 4 ). The OddRoot Property
498 (9 3) Chapter 9 Radicals and Rational Exponents Replace the question mark by an expression that makes the equation correct. Equations involving variables are to be identities. 75. 6 76. 3?? 1 77. 1
More information2.4. Factoring Quadratic Expressions. Goal. Explore 2.4. Launch 2.4
2.4 Factoring Quadratic Epressions Goal Use the area model and Distributive Property to rewrite an epression that is in epanded form into an equivalent epression in factored form The area of a rectangle
More informationApplication. Outline. 31 Polynomial Functions 32 Finding Rational Zeros of. Polynomial. 33 Approximating Real Zeros of.
Polynomial and Rational Functions Outline 31 Polynomial Functions 32 Finding Rational Zeros of Polynomials 33 Approximating Real Zeros of Polynomials 34 Rational Functions Chapter 3 Group Activity:
More informationof surface, 569571, 576577, 578581 of triangle, 548 Associative Property of addition, 12, 331 of multiplication, 18, 433
Absolute Value and arithmetic, 730733 defined, 730 Acute angle, 477 Acute triangle, 497 Addend, 12 Addition associative property of, (see Commutative Property) carrying in, 11, 92 commutative property
More informationChapter 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 informationThe small increase in x is. and the corresponding increase in y is. Therefore
Differentials For a while now, we have been using the notation dy to mean the derivative of y with respect to. Here is any variable, and y is a variable whose value depends on. One of the reasons that
More informationFactoring Polynomials
Factoring Polynomials Hoste, Miller, Murieka September 12, 2011 1 Factoring In the previous section, we discussed how to determine the product of two or more terms. Consider, for instance, the equations
More informationProperties of Real Numbers
16 Chapter P Prerequisites P.2 Properties of Real Numbers What you should learn: Identify and use the basic properties of real numbers Develop and use additional properties of real numbers Why you should
More information6.4 Special Factoring Rules
6.4 Special Factoring Rules OBJECTIVES 1 Factor a difference of squares. 2 Factor a perfect square trinomial. 3 Factor a difference of cubes. 4 Factor a sum of cubes. By reversing the rules for multiplication
More informationThis 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 informationCPM Educational Program
CPM Educational Program A California, NonProfit Corporation Chris Mikles, National Director (888) 8084276 email: mikles @cpm.org CPM Courses and Their Core Threads Each course is built around a few
More informationDefinition 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 informationFSCJ PERT. Florida State College at Jacksonville. assessment. and Certification Centers
FSCJ Florida State College at Jacksonville Assessment and Certification Centers PERT Postsecondary Education Readiness Test Study Guide for Mathematics Note: Pages through are a basic review. Pages forward
More informationSimplification 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 informationWelcome to Math 19500 Video Lessons. Stanley Ocken. Department of Mathematics The City College of New York Fall 2013
Welcome to Math 19500 Video Lessons Prof. Department of Mathematics The City College of New York Fall 2013 An important feature of the following Beamer slide presentations is that you, the reader, move
More information