Section P.3 Algebraic Expressions 25 P.3 Algebraic Expressions What you should learn: Algebraic Expressions •How to identify the terms and coefficients of algebraic A basic characteristic of algebra is the use of letters (or combinations of letters) expressions to represent numbers. The letters used to represent the numbers are variables, and •How to identify the properties of combinations of letters and numbers are algebraic expressions. Here are a few algebra examples. •How to apply the properties of exponents to simplify algebraic x 3x, x ϩ 2, , 2x Ϫ 3y expressions x2 ϩ 1 •How to simplify algebraic expressions by combining like terms and removing symbols Definition of Algebraic Expression of grouping A collection of letters (called variables) and real numbers (called constants) •How to evaluate algebraic expressions combined using the operations of addition, subtraction, multiplication, divi- sion and exponentiation is called an algebraic expression. Why you should learn it: Algebraic expressions can help The terms of an algebraic expression are those parts that are separated by you construct tables of values. addition. For example, the algebraic expression x2 Ϫ 3x ϩ 6 has three terms: x2, For instance, in Example 14 on Ϫ Ϫ page 33, you can determine the 3x, and 6. Note that 3x is a term, rather than 3x, because hourly wages of miners using an x2 Ϫ 3x ϩ 6 ϭ x2 ϩ ͑Ϫ3x͒ ϩ 6. Think of subtraction as a form of addition. expression and a table of values. The terms x2 and Ϫ3x are called the variable terms of the expression, and 6 is called the constant term of the expression. The numerical factor of a variable Study Tip term is called the coefficient of the variable term. For instance, the coefficient of the variable term Ϫ3x is Ϫ3, and the coefficient of the variable term x2 is 1. (The It is important to understand constant term of an expression is also considered to be a coefficient.) the difference between a term and a factor. Terms are separated by addition whereas Example 1 ᭿ Identifying Terms and Coefficients factors are separated by multi- plication. For instance, the Identify the terms and coefficients in each algebraic expression. ͑ ϩ ͒ expression 4x x 2 has 1 1 three factors: 4, x, and (a)5x Ϫ (b)4y ϩ 6x Ϫ 9 (c) x2y Ϫ ϩ 3y 3 x ͑x ϩ 2͒. Solution Terms Coefficients 1 1 (a) 5x, Ϫ 5, Ϫ 3 3 (b)4y, 6x, Ϫ9 4, 6, Ϫ9 1 (c)x2y, Ϫ , 3y 1,Ϫ1, 3 x 26 Chapter P Prerequisites Properties of Algebra The properties of real numbers (see Section P.2) can be used to rewrite algebraic expressions. The following list is similar to those given in Section P.2, except that Historical Note the examples involve algebraic expressions. In other words, the properties are true The French mathematician François for variables and algebraic expressions as well as for real numbers. Viète (1540–1603) was the first to use letters to represent numbers. Properties of Algebra He used vowels to represent Let a, b, and c be real numbers, variables, or algebraic expressions. unknown quantities and conso- nants to represent known Property Example quantities. Commutative Property of Addition a ϩ b ϭ b ϩ a 5x ϩ x2 ϭ x2 ϩ 5x Commutative Property of Multiplication ab ϭ ba ͑3 ϩ x͒x3 ϭ x3͑3 ϩ x͒ Associative Property of Addition ͑a ϩ b͒ ϩ c ϭ a ϩ ͑b ϩ c͒ ͑Ϫx ϩ 6͒ ϩ 3x2 ϭϪx ϩ ͑6 ϩ 3x2͒ Associative Property of Multiplication ͑ab͒c ϭ a͑bc͒ ͑5x и 4y͒͑6͒ ϭ ͑5x͒͑4y и 6͒ Distributive Properties a͑b ϩ c͒ ϭ ab ϩ ac 2x͑4 ϩ 3x͒ ϭ 2x и 4 ϩ 2x и 3x ͑a ϩ b͒c ϭ ac ϩ bc ͑y ϩ 6͒y ϭ y и y ϩ 6 и y Additive Identity Property a ϩ 0 ϭ 0 ϩ a ϭ a 4y2 ϩ 0 ϭ 0 ϩ 4y2 ϭ 4y2 Multiplicative Identity Property a и 1 ϭ 1 и a ϭ a ͑Ϫ5x3͒͑1͒ ϭ ͑1͒͑Ϫ5x3͒ ϭϪ5x3 Additive Inverse Property a ϩ ͑Ϫa͒ ϭ 0 4x2 ϩ ͑Ϫ4x2͒ ϭ 0 Multiplicative Inverse Property 1 1 a и ϭ 1, a 0 ͑x2 ϩ 1͒΂ ΃ ϭ 1 a x2 ϩ 1 Because subtraction is defined as “adding the opposite,” the Distributive Property is also true for subtraction. For instance, the “subtraction form” of a͑b ϩ c͒ ϭ ab ϩ ac is a͑b Ϫ c͒ ϭ a͓b ϩ ͑Ϫc͔͒ ϭ ab ϩ a͑Ϫc͒ ϭ ab Ϫ ac. Section P.3 Algebraic Expressions 27 In addition to these properties, the properties of equality, zero, and negation given in Section P.2 are also valid for algebraic expressions. The next example illustrates the use of a variety of these properties. Example 2 ᭿ Identifying the Properties of Algebra Identify the property of algebra illustrated in each statement. (a)͑5x2͒3 ϭ 3͑5x2͒ (b) ͑3x2 ϩ x͒ Ϫ ͑3x2 ϩ x͒ ϭ 0 (c)3x ϩ 3y2 ϭ 3͑x ϩ y2͒ (d) ͑5 ϩ x2͒ ϩ 4x2 ϭ 5 ϩ ͑x2 ϩ 4x2͒ 1 (e)5x и ϭ 1, x 0 (f ) ͑y Ϫ 6͒3 ϩ ͑y Ϫ 6͒y ϭ ͑y Ϫ 6͒͑3 ϩ y͒ 5x Solution (a) This statement illustrates the Commutative Property of Multiplication. In other words, you obtain the same result whether you multiply 5x2 by 3, or 3 by 5x2. (b) This statement illustrates the Additive Inverse Property. In terms of subtrac- tion, this property simply states that when any expression is subtracted from itself the result is zero. (c) This statement illustrates the Distributive Property. In other words, multipli- cation is distributed over addition. (d) This statement illustrates the Associative Property of Addition. In other words, to form the sum 5 ϩ x2 ϩ 4x2 it does not matter whether 5 and x2 are added first or x2 and 4x2 are added first. (e) This statement illustrates the Multiplicative Inverse Property. Note that it is important that x be a nonzero number. If x were zero, the reciprocal of x would be undefined. (f ) This statement illustrates the Distributive Property in reverse order. ab ϩ ac ϭ a͑b ϩ c͒ Distributive Property ͑y Ϫ 6͒3 ϩ ͑y Ϫ 6͒y ϭ ͑y Ϫ 6͒͑3 ϩ y͒ Note in this case that a ϭ y Ϫ 6, b ϭ 3, and c ϭ y. EXPLORATION Discovering Properties of Exponents Write each of the following as a single power of 2. Explain how you obtained your answer. Then generalize your procedure by completing the statement “When you multiply exponential expressions that have the same base, you. .” a.22 и 23 b.24 и 21 c.25 и 22 d.23 и 24 e. 21 и 25 28 Chapter P Prerequisites Historical Note Properties of Exponents Originally, Arabian mathematicians Just as multiplication by a positive integer can be described as repeated addition, used their words for colors to repre- repeated multiplication can be written in what is called exponential form (see sent quantities (cosa, censa, cubo). Section P.1). Let n be a positive integer and let a be a real number. Then the prod- These words were eventually abbre- uct of n factors of a is given by viated to co, ce, cu. René Descartes an ϭ a и a и a . a. a is the base and n is the exponent. (1596–1650) simplified this even further by introducing the symbols n factors x, x2, and x3. When multiplying two exponential expressions that have the same base, you add exponents. To see why this is true, consider the product a3 и a2. Because the first expression represents a и a и a and the second represents a и a, the product of the two expressions represents a и a и a и a и a, as follows. a3 и a2 ϭ ͑a и a и a͒ и ͑a и a͒ ϭ ͑a и a и a и a и a͒ ϭ a3ϩ2 ϭ a5 3 factors 2 factors 5 factors Study Tip Properties of Exponents Let a and b represent real numbers, variables, or algebraic expressions, and The first and second properties let m and n be positive integers. of exponents can be extended to products involving three or Property Example more factors. For example, ϩ ϩ 1. aman ϭ am n x5͑x4͒ ϭ x5 4 ϭ x9 am и an и ak ϭ amϩnϩk 2. ͑ab͒m ϭ ambm ͑2x͒3 ϭ 23͑x3͒ ϭ 8x3 and 3. ͑am͒n ϭ amn ͑x2͒3 ϭ x2и3 ϭ x6 ͑abc͒m ϭ ambmcm. am x6 4. ϭ amϪn, m > n, a 0 ϭ x6Ϫ2 ϭ x4, x 0 an x2 a m am x 3 x3 x3 5. ΂ ΃ ϭ , b 0 ΂ ΃ ϭ ϭ b bm 2 23 8 Example 3 ᭿ Illustrating the Properties of Exponents (a) To multiply exponential expressions that have the same base,add exponents. x2 и x4 ϭ x и x и x и x и x и x ϭ x и x и x и x и x и x ϭ x2ϩ4 ϭ x6 2 factors 4 factors 6 factors (b) To raise the product of two factors to the same power,raise each factor to the power and multiply the results. ͑3x͒3 ϭ 3x и 3x и 3x ϭ 3 и 3 и 3 и x и x и x ϭ 33 и x3 ϭ 27x3 3 factors 3 factors 3 factors (c) To raise an exponential expression to a power, multiply exponents. ͑x3͒2 ϭ ͑x и x и x͒ и ͑x и x и x͒ ϭ ͑x и x и x и x и x и x͒ ϭ x3и2 ϭ x6 3 factors 3 factors 6 factors Section P.3 Algebraic Expressions 29 ᭿ Study Tip Example 4 Illustrating the Properties of Exponents (a) To divide exponential expressions that have the same base, subtract In the expression x ϩ 5, the exponents.