DOCSLIB.ORG

Math 380 – Elementary Algebra

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Math 380 – Elementary Algebra Elementary Algebra Textbook Second Edition Department of Mathematics College of the Redwoods 2012-2013 ii Copyright All parts of this prealgebra textbook are copyrighted c 2011 in the name of the Department of Mathematics, College of the Redwoods. They are not in the public domain. However, they are being made available free for use in educational institutions. This offer does not extend to any application that is made for profit. Users who have such applications in mind should contact David Arnold at [email protected] or Bruce Wagner at [email protected]. This work is licensed under the Creative Commons Attribution-Non- Commercial-NoDerivs 3.0 Unported License. To view a copy of this li- cense, visit http://creativecommons.org/licenses/by-nc-nd/3.0/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Fran- cisco, California, 94105, USA. Contents 1 The Arithmetic of Numbers 1 1.1 An Introduction to the Integers .................. 2 The Integers ........................... 3 Absolute Value .......................... 4 Integer Addition ......................... 5 Mathematical Properties of Addition .............. 6 Integer Subtraction ....................... 7 Integer Multiplication ...................... 8 Mathematical Properties of Multiplication ........... 9 Exponents ............................ 10 Graphing Calculator: Negating versus Subtracting ...... 11 Exercises ............................. 14 Answers .............................. 15 1.2 Order of Operations ........................ 16 Grouping Symbols ........................ 19 Absolute Value Bars as Grouping Symbols .......... 20 Nested Grouping Symbols .................... 20 Evaluating Algebraic Expressions ................ 21 Evaluating Fractions ....................... 22 Using the Graphing Calculator ................. 23 Exercises ............................. 27 Answers .............................. 30 1.3 The Rational Numbers ....................... 31 Reducing Fractions to Lowest Terms .............. 31 Multiplying Fractions ...................... 33 Dividing Fractions ........................ 35 Adding Fractions ......................... 36 Order of Operations ....................... 37 Fractions on the Graphing Calculator ............. 39 Exercises ............................. 43 Answers .............................. 46 iii iv CONTENTS 1.4 Decimal Notation .......................... 48 Adding and Subtracting Decimals ............... 49 Multiplication and Division of Decimals ............ 51 Order of Operations ....................... 52 Rounding Using the Graphing Calculator ........... 55 Exercises ............................. 57 Answers .............................. 59 1.5 Algebraic Expressions ....................... 60 The Distributive Property .................... 61 Speeding Things Up a Bit .................... 62 Distributing a Negative Sign .................. 63 Combining Like Terms ...................... 64 Order of Operations ....................... 66 Exercises ............................. 69 Answers .............................. 70 2 Solving Linear Equations and Inequalities 73 2.1 Solving Equations: One Step ................... 74 Equivalent Equations ...................... 74 Wrap and Unwrap, Do and Undo ................ 75 Operations that Produce Equivalent Equations ........ 76 More Operations That Produce Equivalent Equations .... 79 Writing Mathematics ...................... 82 Exercises ............................. 84 Answers .............................. 85 2.2 Solving Equations: Multiple Steps ................ 87 Variables on Both Sides of the Equation ............ 90 Simplifying Expressions When Solving Equations ....... 92 Exercises ............................. 96 Answers .............................. 97 2.3 Clearing Fractions and Decimals ................. 99 Canceling is More Efficient ................... 100 Clearing Fractions from an Equation .............. 101 Clearing Decimals from an Equation .............. 106 Exercises ............................. 109 Answers .............................. 110 2.4 Formulae .............................. 111 Clearing Fractions ........................ 113 Geometric Formulae ....................... 115 Exercises ............................. 118 Answers .............................. 120 2.5 Applications ............................. 121 Exercises ............................. 132 Answers .............................. 134 2.6 Inequalities ............................. 135 Ordering the Real Numbers ................... 136 CONTENTS v Set-Builder Notation ....................... 136 Interval Notation ......................... 138 Equivalent Inequalities ..................... 139 Reversing the Inequality Sign .................. 141 Multiple Steps .......................... 143 Summary Table of Set-Builder and Interval Notation ..... 146 Exercises ............................. 147 Answers .............................. 149 3 Introduction to Graphing 151 3.1 Graphing Equations by Hand ................... 152 The Cartesian Coordinate System ............... 152 Plotting Ordered Pairs ..................... 154 Equations in Two Variables ................... 156 Graphing Equations in Two Variables ............. 157 Guidelines and Requirements .................. 159 Using the TABLE Feature of the Graphing Calculator .... 160 Exercises ............................. 165 Answers .............................. 168 3.2 The Graphing Calculator ..................... 171 Reproducing Calculator Results on Homework Paper .... 174 Adjusting the Viewing Window ................. 176 Exercises ............................. 179 Answers .............................. 180 3.3 Rates and Slope .......................... 183 Measuring the Change in a Variable .............. 184 Slope as Rate ........................... 185 The Steepness of a Line ..................... 189 The Geometry of the Slope of a Line .............. 192 Exercises ............................. 196 Answers .............................. 199 3.4 Slope-Intercept Form of a Line .................. 201 Applications ........................... 206 Exercises ............................. 209 Answers .............................. 212 3.5 Point-Slope Form of a Line .................... 213 Parallel Lines ........................... 217 Perpendicular Lines ....................... 219 Applications ........................... 222 Exercises ............................. 224 Answers .............................. 225 3.6 Standard Form of a Line ...................... 228 Slope-Intercept to Standard Form ............... 230 Point-Slope to Standard Form ................. 232 Intercepts ............................. 234 Horizontal and Vertical Lines .................. 237 vi CONTENTS Exercises ............................. 240 Answers .............................. 242 4 Systems of Linear Equations 245 4.1 Solving Systems by Graphing ................... 246 Exceptional Cases ........................ 250 Solving Systems with the Graphing Calculator ........ 253 Exercises ............................. 259 Answers .............................. 260 4.2 Solving Systems by Substitution ................. 262 Exceptional Cases Revisited .................. 268 Exercises ............................. 270 Answers .............................. 272 4.3 Solving Systems by Elimination .................. 273 Exceptional Cases ........................ 276 Exercises ............................. 279 Answers .............................. 280 4.4 Applications of Linear Systems .................. 282 Exercises ............................. 290 Answers .............................. 291 5 Polynomial Functions 293 5.1 Functions .............................. 294 Mapping Diagrams ........................ 296 Function Definition ....................... 297 Mapping Diagram Notation ................... 299 Function Notation ........................ 301 Interchanging y and f(x) .................... 303 Exercises ............................. 305 Answers .............................. 307 5.2 Polynomials ............................. 308 Ascending and Descending Powers ............... 309 The Degree of a Polynomial ................... 311 Polynomial Functions ...................... 312 The Graph of a Polynomial Function ............. 313 Exercises ............................. 317 Answers .............................. 319 5.3 Applications of Polynomials .................... 321 Zeros and x-intercepts of a Function .............. 325 Exercises ............................. 330 Answers .............................. 333 5.4 Adding and Subtracting Polynomials ............... 334 Negating a Polynomial ..................... 335 Subtracting Polynomials ..................... 336 Some Applications ........................ 338 Exercises ............................. 340 CONTENTS vii Answers .............................. 342 5.5 Laws of Exponents ......................... 343 Multiplying With Like Bases .................. 344 Dividing With Like Bases .................... 345 Raising a Power to a Power ................... 347 Raising a Product to a Power .................. 349 Raising a Quotient to a Power ................. 350 Exercises ............................. 353 Answers .............................. 355 5.6 Multiplying Polynomials ...................... 356 The Product of Monomials ................... 356 Multiplying a Monomial and a Polynomial .......... 357 Multiplying Polynomials ..................... 359 Speeding Things Up a Bit ...................
Load more

Recommended publications
  • 1. Make Sense of Problems and Persevere in Solving Them. Mathematically Proficient Students Start by Explaining to Themselves T
    1. Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. 2. Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize —to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize , to pause as needed during the manipulation process in order to probe into the referents for the symbols involved.
    [Show full text]
  • Higher Mathematics
    ; HIGHER MATHEMATICS Chapter I. THE SOLUTION OF EQUATIONS. By Mansfield Merriman, Professor of Civil Engineering in Lehigh University. Art. 1. Introduction. In this Chapter will be presented a brief outline of methods, not commonly found in text-books, for the solution of an equation containing one unknown quantity. Graphic, numeric, and algebraic solutions will be given by which the real roots of both algebraic and transcendental equations may be ob- tained, together with historical information and theoretic discussions. An algebraic equation is one that involves only the opera- tions of arithmetic. It is to be first freed from radicals so as to make the exponents of the unknown quantity all integers the degree of the equation is then indicated by the highest ex- ponent of the unknown quantity. The algebraic solution of an algebraic equation is the expression of its roots in terms of literal is the coefficients ; this possible, in general, only for linear, quadratic, cubic, and quartic equations, that is, for equations of the first, second, third, and fourth degrees. A numerical equation is an algebraic equation having all its coefficients real numbers, either positive or negative. For the four degrees 2 THE SOLUTION OF EQUATIONS. [CHAP. I. above mentioned the roots of numerical equations may be computed from the formulas for the algebraic solutions, unless they fall under the so-called irreducible case wherein real quantities are expressed in imaginary forms. An algebraic equation of the n th degree may be written with all its terms transposed to the first member, thus: n- 1 2 x" a x a,x"- .
    [Show full text]
  • Arxiv:2008.02889V1 [Math.QA]
    NONCOMMUTATIVE NETWORKS ON A CYLINDER S. ARTHAMONOV, N. OVENHOUSE, AND M. SHAPIRO Abstract. In this paper a double quasi Poisson bracket in the sense of Van den Bergh is constructed on the space of noncommutative weights of arcs of a directed graph embedded in a disk or cylinder Σ, which gives rise to the quasi Poisson bracket of G.Massuyeau and V.Turaev on the group algebra kπ1(Σ,p) of the fundamental group of a surface based at p ∈ ∂Σ. This bracket also induces a noncommutative Goldman Poisson bracket on the cyclic space C♮, which is a k-linear space of unbased loops. We show that the induced double quasi Poisson bracket between boundary measurements can be described via noncommutative r-matrix formalism. This gives a more conceptual proof of the result of [Ove20] that traces of powers of Lax operator form an infinite collection of noncommutative Hamiltonians in involution with respect to noncommutative Goldman bracket on C♮. 1. Introduction The current manuscript is obtained as a continuation of papers [Ove20, FK09, DF15, BR11] where the authors develop noncommutative generalizations of discrete completely integrable dynamical systems and [BR18] where a large class of noncommutative cluster algebras was constructed. Cluster algebras were introduced in [FZ02] by S.Fomin and A.Zelevisnky in an effort to describe the (dual) canonical basis of universal enveloping algebra U(b), where b is a Borel subalgebra of a simple complex Lie algebra g. Cluster algebras are commutative rings of a special type, equipped with a distinguished set of generators (cluster variables) subdivided into overlapping subsets (clusters) of the same cardinality subject to certain polynomial relations (cluster transformations).
    [Show full text]
  • Using March Madness in the First Linear Algebra Course
    Using March Madness in the first Linear Algebra course Steve Hilbert Ithaca College [email protected] Background • National meetings • Tim Chartier • 1 hr talk and special session on rankings • Try something new Why use this application? • This is an example that many students are aware of and some are interested in. • Interests a different subgroup of the class than usual applications • Interests other students (the class can talk about this with their non math friends) • A problem that students have “intuition” about that can be translated into Mathematical ideas • Outside grading system and enforcer of deadlines (Brackets “lock” at set time.) How it fits into Linear Algebra • Lots of “examples” of ranking in linear algebra texts but not many are realistic to students. • This was a good way to introduce and work with matrix algebra. • Using matrix algebra you can easily scale up to work with relatively large systems. Filling out your bracket • You have to pick a winner for each game • You can do this any way you want • Some people use their “ knowledge” • I know Duke is better than Florida, or Syracuse lost a lot of games at the end of the season so they will probably lose early in the tournament • Some people pick their favorite schools, others like the mascots, the uniforms, the team tattoos… Why rank teams? • If two teams are going to play a game ,the team with the higher rank (#1 is higher than #2) should win. • If there are a limited number of openings in a tournament, teams with higher rankings should be chosen over teams with lower rankings.
    [Show full text]
  • Ring (Mathematics) 1 Ring (Mathematics)
    Ring (mathematics) 1 Ring (mathematics) In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition (called the additive group of the ring) and a monoid under multiplication such that multiplication distributes over addition.a[›] In other words the ring axioms require that addition is commutative, addition and multiplication are associative, multiplication distributes over addition, each element in the set has an additive inverse, and there exists an additive identity. One of the most common examples of a ring is the set of integers endowed with its natural operations of addition and multiplication. Certain variations of the definition of a ring are sometimes employed, and these are outlined later in the article. Polynomials, represented here by curves, form a ring under addition The branch of mathematics that studies rings is known and multiplication. as ring theory. Ring theorists study properties common to both familiar mathematical structures such as integers and polynomials, and to the many less well-known mathematical structures that also satisfy the axioms of ring theory. The ubiquity of rings makes them a central organizing principle of contemporary mathematics.[1] Ring theory may be used to understand fundamental physical laws, such as those underlying special relativity and symmetry phenomena in molecular chemistry. The concept of a ring first arose from attempts to prove Fermat's last theorem, starting with Richard Dedekind in the 1880s. After contributions from other fields, mainly number theory, the ring notion was generalized and firmly established during the 1920s by Emmy Noether and Wolfgang Krull.[2] Modern ring theory—a very active mathematical discipline—studies rings in their own right.
    [Show full text]
  • Section X.56. Insolvability of the Quintic
    X.56 Insolvability of the Quintic 1 Section X.56. Insolvability of the Quintic Note. Now is a good time to reread the first set of notes “Why the Hell Am I in This Class?” As we have claimed, there is the quadratic formula to solve all polynomial equations ax2 +bx+c = 0 (in C, say), there is a cubic equation to solve ax3+bx2+cx+d = 0, and there is a quartic equation to solve ax4+bx3+cx2+dx+e = 0. However, there is not a general algebraic equation which solves the quintic ax5 + bx4 + cx3 + dx2 + ex + f = 0. We now have the equipment to establish this “insolvability of the quintic,” as well as a way to classify which polynomial equations can be solved algebraically (that is, using a finite sequence of operations of addition [or subtraction], multiplication [or division], and taking of roots [or raising to whole number powers]) in a field F . Definition 56.1. An extension field K of a field F is an extension of F by radicals if there are elements α1, α2,...,αr ∈ K and positive integers n1,n2,...,nr such that n1 ni K = F (α1, α2,...,αr), where α1 ∈ F and αi ∈ F (α1, α2,...,αi−1) for 1 <i ≤ r. A polynomial f(x) ∈ F [x] is solvable by radicals over F if the splitting field E of f(x) over F is contained in an extension of F by radicals. n1 Note. The idea in this definition is that F (α1) includes the n1th root of α1 ∈ F .
    [Show full text]
  • The Notion Of" Unimaginable Numbers" in Computational Number Theory
    Beyond Knuth’s notation for “Unimaginable Numbers” within computational number theory Antonino Leonardis1 - Gianfranco d’Atri2 - Fabio Caldarola3 1 Department of Mathematics and Computer Science, University of Calabria Arcavacata di Rende, Italy e-mail: [email protected] 2 Department of Mathematics and Computer Science, University of Calabria Arcavacata di Rende, Italy 3 Department of Mathematics and Computer Science, University of Calabria Arcavacata di Rende, Italy e-mail: [email protected] Abstract Literature considers under the name unimaginable numbers any positive in- teger going beyond any physical application, with this being more of a vague description of what we are talking about rather than an actual mathemati- cal definition (it is indeed used in many sources without a proper definition). This simply means that research in this topic must always consider shortened representations, usually involving recursion, to even being able to describe such numbers. One of the most known methodologies to conceive such numbers is using hyper-operations, that is a sequence of binary functions defined recursively starting from the usual chain: addition - multiplication - exponentiation. arXiv:1901.05372v2 [cs.LO] 12 Mar 2019 The most important notations to represent such hyper-operations have been considered by Knuth, Goodstein, Ackermann and Conway as described in this work’s introduction. Within this work we will give an axiomatic setup for this topic, and then try to find on one hand other ways to represent unimaginable numbers, as well as on the other hand applications to computer science, where the algorith- mic nature of representations and the increased computation capabilities of 1 computers give the perfect field to develop further the topic, exploring some possibilities to effectively operate with such big numbers.
    [Show full text]
  • Interval Notation (Mixed Parentheses and Brackets) for Interval Notation Using Parentheses on Both Sides Or Brackets on Both Sides, You Can Use the Basic View
    Canvas Equation Editor Tips: Advanced View With the Advanced View of the Canvas Equation Editor, you can input more complicated mathematical text using LaTeX, like matrices, interval notation, and piecewise functions. In case you don’t have a lot of LaTeX experience, here are some tips for how to write LaTex for some commonly used mathematics. More tips can be found in the Basic View Tips PDF. ​ ​ ​ Interval Notation (mixed parentheses and brackets) For interval notation using parentheses on both sides or brackets on both sides, you can use the basic view. Just type ( to get ( ) or [ to get [ ] in the editor. Left parenthesis \left( infinity symbol \infty Left bracket \left[ negative infinity -\infty Right parenthesis \right) square root \sqrt{ } Right bracket \right] fraction \frac{ }{ } Below are several examples that should help you construct the LaTeX for the interval notation you need. Display LaTeX in Advanced View \left(x,y\right] \left[x,y\right) \left[-2,\infty\right) \left(-\infty,4\right] \left(\sqrt{5} ,\frac{3}{4}\right] © Canvas 2021 | guides.canvaslms.com | updated 2017-12-09 Page 1 Canvas Equation Editor Tips: Advanced View Piecewise Functions You can use the Advanced View of the Canvas Equation Editor to write piecewise functions. To do this, you will need to write the appropriate LaTeX to express the mathematical text (see example below): This kind of LaTeX expression is like an onion - there are many layers. The outer layer declares the function and a resizing brace: The next layer in builds an array to hold the function definitions and conditions: The innermost layer defines the functions and conditions.
    [Show full text]
  • Redundant Cloud Services Skyrocketing! How to Provision Minecraft Server on Metacloud
    Cisco Service Provider Cloud Josip Zimet CCIE 5688 Cisco My Favorite Example of Digital Transformations started with data centers …. Example of Digital Transformations started with data centers …. Paris Dubai Dubrovnik https://developer.cisco.com/site/flare/ SIM Card Identity for a Phone + SIM Card Identity for a Phone HSRP x.x.x.1 + STP/802.1Q/FP IS-IS/BGP/VXLAN Anycast GW x.x.x.1 Physical, virtual, Container SIM Card Identity for a Phone + Multitenant multivendor across bare metal, virtual and container private and public cloud Cloud Center VMs=house Apartments=containers Nova/cinder/Neutron NSX/ACI/Contiv EC2/S3/EBS/VPC/Sec Groups Broad Multi-Vendor Infrastructure Support UCS Director Converged VM L4-L7 Compute Network Storage vASA, Nexus CSR1000v MDS * * * * * * * * * Partner provided roleback https://www.youtube.com/watch?v=hz7zwd98rn4 No Web-1 No Web-2 No App-1 No DB-1 No DB-2 No DB-3 No App-2 No DB-3 Value of Sec ? st 0.36 Seconds nd 1 Place ($881,000) separates 2 Place 1st and 2nd place • $2,447 Per Millisecond • $1.6 more dollars awarded to • $719,000 Million 1st Place Value of Sec ? Financial Media Transport Retail Airline Brokerage Home Shopping Pay per View Reservations operations $1,883/min $2,500/min $1,483/min $107,500/min Credit card/Sales Authorizations Teleticket Package Catalogue Sales $43,333/min Sales Shipping $1,500/min $1,150/min $466/min ATM Fees $241/min Classification Availability Annual Down Time Continuous processing 100% 0 min/year Fault Tolerant 99-999% 5 min/Year Fault Resilient 99.99% 53 min/year High Availability
    [Show full text]
  • Fundamental Theorems in Mathematics
    SOME FUNDAMENTAL THEOREMS IN MATHEMATICS OLIVER KNILL Abstract. An expository hitchhikers guide to some theorems in mathematics. Criteria for the current list of 243 theorems are whether the result can be formulated elegantly, whether it is beautiful or useful and whether it could serve as a guide [6] without leading to panic. The order is not a ranking but ordered along a time-line when things were writ- ten down. Since [556] stated “a mathematical theorem only becomes beautiful if presented as a crown jewel within a context" we try sometimes to give some context. Of course, any such list of theorems is a matter of personal preferences, taste and limitations. The num- ber of theorems is arbitrary, the initial obvious goal was 42 but that number got eventually surpassed as it is hard to stop, once started. As a compensation, there are 42 “tweetable" theorems with included proofs. More comments on the choice of the theorems is included in an epilogue. For literature on general mathematics, see [193, 189, 29, 235, 254, 619, 412, 138], for history [217, 625, 376, 73, 46, 208, 379, 365, 690, 113, 618, 79, 259, 341], for popular, beautiful or elegant things [12, 529, 201, 182, 17, 672, 673, 44, 204, 190, 245, 446, 616, 303, 201, 2, 127, 146, 128, 502, 261, 172]. For comprehensive overviews in large parts of math- ematics, [74, 165, 166, 51, 593] or predictions on developments [47]. For reflections about mathematics in general [145, 455, 45, 306, 439, 99, 561]. Encyclopedic source examples are [188, 705, 670, 102, 192, 152, 221, 191, 111, 635].
    [Show full text]
  • Matrices Lie: an Introduction to Matrix Lie Groups and Matrix Lie Algebras
    Matrices Lie: An introduction to matrix Lie groups and matrix Lie algebras By Max Lloyd A Journal submitted in partial fulfillment of the requirements for graduation in Mathematics. Abstract: This paper is an introduction to Lie theory and matrix Lie groups. In working with familiar transformations on real, complex and quaternion vector spaces this paper will define many well studied matrix Lie groups and their associated Lie algebras. In doing so it will introduce the types of vectors being transformed, types of transformations, what groups of these transformations look like, tangent spaces of specific groups and the structure of their Lie algebras. Whitman College 2015 1 Contents 1 Acknowledgments 3 2 Introduction 3 3 Types of Numbers and Their Representations 3 3.1 Real (R)................................4 3.2 Complex (C).............................4 3.3 Quaternion (H)............................5 4 Transformations and General Geometric Groups 8 4.1 Linear Transformations . .8 4.2 Geometric Matrix Groups . .9 4.3 Defining SO(2)............................9 5 Conditions for Matrix Elements of General Geometric Groups 11 5.1 SO(n) and O(n)........................... 11 5.2 U(n) and SU(n)........................... 14 5.3 Sp(n)................................. 16 6 Tangent Spaces and Lie Algebras 18 6.1 Introductions . 18 6.1.1 Tangent Space of SO(2) . 18 6.1.2 Formal Definition of the Tangent Space . 18 6.1.3 Tangent space of Sp(1) and introduction to Lie Algebras . 19 6.2 Tangent Vectors of O(n), U(n) and Sp(n)............. 21 6.3 Tangent Space and Lie algebra of SO(n).............. 22 6.4 Tangent Space and Lie algebras of U(n), SU(n) and Sp(n)..
    [Show full text]
  • 8 Algebra: Bracketsmep Y8 Practice Book A
    8 Algebra: BracketsMEP Y8 Practice Book A 8.1 Expansion of Single Brackets In this section we consider how to expand (multiply out) brackets to give two or more terms, as shown below: 36318()xx+=+ First we revise negative numbers and order of operations. Example 1 Evaluate: (a) −+610 (b) −+−74() (c) ()−65×−() (d) 647×−() (e) 48()+ 3 (f) 68()− 15 ()−23−−() (g) 35−−() (h) −1 Solution (a) −+6104 = (b) −+−7474()=− − =−11 (c) ()−6530×−()= (d) 6476×−()=×−() 3 =−18 (e) 48()+ 3=× 4 11 = 44 (f) 68()− 15=×− 6() 7 =−42 127 8.1 MEP Y8 Practice Book A (g) 3535−−()=+ = 8 ()−23−−()()−23+ (h) = −1 −1 1 = −1 =−1 When a bracket is expanded, every term inside the bracket must be multiplied by the number outside the bracket. Remember to think about whether each number is positive or negative! Example 2 Expand 36()x+ using a table. Solution × x 6 3 3x 18 From the table, 36318()xx+=+ Example 3 Expand 47()x−. Solution 47()x−=×−×447x Remember that every term inside the bracket must be multiplied by =−428x the number outside the bracket. Example 4 Expand xx()8−. 128 MEP Y8 Practice Book A Solution xx()8−=×−×xxx8 =−8xx2 Example 5 Expand ()−34()− 2x. Solution ()−34()− 2x=−()34×−−() 32×x =−12 −() − 6 x =−12 + 6 x Exercises 1. Calculate: (a) −+617 (b) 614− (c) −−65 (d) 69−−() (e) −−−11() 4 (f) ()−64×−() (g) 87×−() (h) 88÷−() 4 (i) 68()− 10 (j) 53()− 10 (k) 711()− 4 (l) ()− 4617()− 2. Copy and complete the following tables, and write down each of the expansions: (a) (b) ×x 2 ×x – 7 4 5 42()x+= 57()x−= (c) (d) × x 3 ×2x 5 4 5 43()x+= 52()x+ 5= 129 8.1 MEP Y8 Practice Book A 3.
    [Show full text]