Hawkes Learning Systems: College Algebra

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1 Hawkes Learning Systems: College Algebra Section 1.2: The Arithmetic of Algebraic Expressions

2 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.

3 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 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.

4 Example: Components and Terminology of an Algebraic Expression Consider the algebraic expression What are its terms? What is the coefficient of each of the terms? 3, -2, 7, -12 What is the variable factor of each of the terms? (x y), none

5 Evaluating Algebraic Expressions o To evaluate an expression means to replace the variables with constants, perform the mathematical operations and simplify.

6 Example: Evaluate an Algebraic Expression Evaluate the following algebraic expression: 2 2x 5( x y) 2 2x 5( x y) for x 1and y 4 Replace x with 1 and y with 4. Simplify. 27

7 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 Closure Commutative Associative Identity Inverse Distributive a Additive Version b is real number Multiplicative Version is a real number a b b a ab ba a ( b c) ( a b) c a 0 0 a a a ( a) 0 ab a( b c) ab ac a( bc) ( ab) c a 1 1 a a 1 a 1(for a 0) a

8 Example: Field Properties Identify the property that justifies the statement. a) 4(y 3) = 4y 12 b) -3(4x 6 z) = (-3 4)(x 6 z) = -12x 6 z c)

9 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 A B A C B C For C 0, A B A C B C Description 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

10 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 AB 0 A 0 or B 0 for implies, we write

11 Example: Properties Identify the property that justifies the statement. (If it s a cancellation property, identify the quantity added or multiplied on both sides.) a) -14y = 7 y = -½ b) (x 3)(x + 2) = 0 x 3 = 0 or x + 2 = 0 c)

12 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.

13 Example: Order of Operations Evaluate following Order of Operations. a) b)

14 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.

15 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, if it is in the set A, A B the set B, or both. Note: the union of A and B contains both individual sets.

16 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 is, an element x is in A B, A B if it is in both A and B. Note: the intersection of A and B is contained in each individual set.

17 Example: Set Operations Simplify each of the following set expressions. a. b. c. d. 3,7 1,11 3,11 3,7 1,11 1,7 2,5 5,7,2 17,, Since these two intervals overlap, their union is best described with a single interval. This intersection of two intervals can also be described with a single interval. These two intervals have no elements in common so their intersection is the empty set. The union of these two intervals constitutes the entire set of real numbers.

18 Example: Set Operations Simplify each of the following set expressions. a) b)

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