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HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Hawkes Learning Systems: College Algebra Section 1.2: The Arithmetic of Algebraic Expressions HAWKES LEARNING SYSTEMS Copyright © 2010 Hawkes Learning Systems. All rights reserved. math courseware specialists Objectives o Components and terminology of algebraic expressions. o The field properties and their use in algebra. o The order of mathematical operations. o Basic set operations and Venn diagrams. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Components and Terminology of Algebraic Expressions o Algebraic Expressions are made of constants and variables combined by mathematical operations. o Constants are fixed numbers. o Variables are unspecified numbers. o To evaluate an expression means to replace the variables with constants, perform the mathematical operations and simplify. o Terms are the parts of an algebraic expression joined by addition (or subtraction). o Factors of a term are the parts of a term that are joined by multiplication (or division). o The coefficient of a term is the constant factor of the term while the remaining part of the term constitutes the variable factor. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 1: Components and Terminology of Algebraic Expressions Consider the algebraic expression 13 x 4 ( x 3 5 y ) 7 x 19( x y ) What are its terms? 4 3 13 x ( x 5 y ), 7 x , 19( x y ) What is the coefficient of each of the terms? 13, 7, 19 What is the variable factor of each of the terms? x 4 ( x 3 5 y ), x , ( x y ) HAWKES LEARNING SYSTEMS Copyright © 2010 Hawkes Learning Systems. All rights reserved. math courseware specialists Example 2: Components and Terminology of Algebraic Expressions Evaluate the following algebraic expression: 2 x 2 5( x y ) for x 1 and y 4 2 x 2 5( x y ) 2(1)2 5 (1) (4) Replace x with 1 and y with 4. 2(1) 5(5) Simplify. 2 25 27 HAWKES LEARNING SYSTEMS Copyright © 2010 Hawkes Learning Systems. All rights reserved. math courseware specialists The Field Properties and Their Use in Algebra The set of real numbers forms what is known mathematically as a field and the properties below are called field properties. a, b and c represent arbitrary real numbers. Name of Property Additive Version Closure a b is real number ab is a real number Commutative ab ba ab ba Associative a (b c ) (a b ) c a (bc ) (ab)c Identity a 0 0 a a Inverse Distributive Multiplicative Version a 1 1 a a 1 a 1 (for a 0) a ( a ) 0 a a (b c ) ab ac HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 3: The Field Properties and Their Use in Algebra is the 2 4 2 ( 1) a. 2 4 (1) This same as 8 2 2 3 6 6 This is the 2 3 b. 3 2 same as 3 (2) 1 1 1 3 3 , x 2y 0 c. ( x 2 y ) 3 x 2y x3 2 y 3 1 x 2y These two algebraic expressions are the same because of the distributive property. These two algebraic expressions are the same because of the commutative property of addition. Any non-zero expression, when multiplied by its reciprocal, yields the multiplicative identity, 1. 1 Here, x 3 2 y is the multiplicative inverse of x 3 2 y. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. The Field Properties and Their Use in Algebra Cancellation Properties Throughout this table, A, B and C represent algebraic expressions. The symbol can be read as “if and only if” or “is equivalent to.” Property Description A B AC B C For C 0, A B AC B C Additive Cancellation Adding the same quantity to both sides of an equation results in an equivalent equation Multiplicative Cancellation Multiplying both sides of an equation by the same non-zero quantity results in an equivalent equation HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. The Field Properties and Their Use in Algebra Zero-Factor Property Let A and B represent algebraic expressions. If the product of A and B is 0, then at least one of A and B is itself 0. Using the symbol for “implies,” we write AB 0 A 0 or B 0 HAWKES LEARNING SYSTEMS Copyright © 2010 Hawkes Learning Systems. All rights reserved. math courseware specialists Example 4: The Field Properties and Their Use in Algebra Verify the equivalence of the following algebraic expressions. Remember that means “is a. 5 x 2 y 2 z z 3 2 z 5 x 2 y z 3 equivalent to.” Isthe equivalence 5 x 2 y 2z z 3 2z true? Use additive cancellation by adding 2z 2z 2z to both sides of the equation. 5x 2 y z 3 5x 2 y z 3 2 2 y y b. 5 x 10 x 2 4 4 2 1 1 y 5 x 10 5 5 4 2 2 y y x 2 x 2 4 4 We can see that the equivalence is true. Is the equivalence true? Use multiplicative cancellation by multiplying both sides of the 1 equation by . 5 We can see that the equivalence is true. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 5: The Field Properties and Their Use in Algebra Verify the following algebraic expressions. c. xy 14 0 0 Is the equivalence true? No. While multiplying both sides of the equation by 0 0 xy 14 0 does result in 0=0, the two expressions are NOT 0 0 0 0 equivalent. Recall that the multiplicative cancellation property requires that the number multiplied by both sides is non-zero. d. x y x y 0 x y 0 or x y 0 Remember that the symbol means “implies.” Is the above implication true? Yes. Recall the zero-factor property. It states that if the product of two terms is 0, then at least one of the terms must be 0. Thus, x – y = 0 or x + y = 0. In mathematics, “or” is not exclusive, so it could be true that both factors equal 0. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. The Order of Mathematical Operations Order of Operations 1. If the expression is a fraction, simplify the numerator and denominator individually, according to the guidelines in the following steps. 2. Parentheses, braces and brackets are all used as grouping symbols. Simplify expressions within each set of grouping symbols, if any are present, working from the innermost outward. 3. Simplify all powers (exponents) and roots. 4. Perform all multiplications and divisions in the expression in the order they occur, working from left to right. 5. Perform all additions and subtractions in the expression in the order they occur, working from left to right. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 6: The Order of Mathematical Operations Follow the order of operations to solve the following. 14 The first step is to simplify the fraction. a.12 7 Next, we subtract. 12 2 10 b. 22 First we must simplify the power, 22. Notice that this can be written as 1 times 22. Simplify the 2 1 2 power. Multiply. 1 4 2 4 Note: What if the expression was 2 ? Then we would square 2 instead of 2 and the answer would be 4. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 7: The Order of Mathematical Operations Follow the order of operations to solve the following: First, we simplify the numerator and 2 8 5 2 9 1 denominator separately. In the numerator, 2 first simplify the roots and powers and then simplify the fractions. In the denominator 10 5 1 simplify the multiplication. 5 2 3 1 4 In the numerator, simplify inside the parenthesis first. In the denominator, subtract 10 (5) 5 from 10. In the numerator, remember to multiply 5 (5)(4) before subtracting. 5 5 20 5 5 HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Basic Set Operations and Venn Diagrams o Set operations, union and intersection, combine two or more sets. o A Venn diagram is a pictorial representation of a set or sets and its aim is to indicate, through shading, the outcome of set operations such as union and intersection. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Basic Set Operations and Venn Diagrams A and B denote two sets and are represented in the Venn diagram by circles. The operation of union is demonstrated by shading. The symbol is read “is an element of”. The union of A and B, denoted A B, is the set x | x A or x B. That is, an element x is in A B A B, if it is in the set A, the set B, or both. Note: the union of A and B contains both individual sets. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Basic Set Operations and Venn Diagrams A and B denote two sets and are represented in the Venn diagram by circles. The operation of intersection is demonstrated by shading. The symbol is read “is an element of”. The intersection of A and B, denoted A B, is the set x | x A and x B. That A B is, an element x is in A B, if it is in both A and B. Note: the intersection of A and B is contained in each individual set. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 8: Basic Set Operations and Venn Diagrams Simplify each of the following set expressions. a. 3,7 1,11 Since these two intervals overlap, their union is best described with a single interval. 3,11 b. 3,7 1,11 This intersection of two intervals can also be described with a single interval. 1,7 These two intervals have no elements in c. 2,5 5,7 common so their intersection is the empty set. d. ,2 17, The union of these two intervals constitutes the entire set of real numbers. ,