Math 017 Materials With Exercises

Lesson 3__________________________________________________________________________________Topics: Equivalent algebraic expressions.__________________________________________________________________________________Definition of equivalent algebraic expressionsSuppose we wish to write as an algebraic expression “a number x doubled”. Should we write x 2 or2 x ? Because of the commutative property of multiplication, both answers are right. Both have thesame meaning, although they appear to be different. We encounter a similar idea in arithmetic. The2 1fractions and are equivalent, which means that they represent the same number although they „do4 2not look the same‟. Similarly, we would say that x 2 and 2 x are equivalent (we often say equal) andwrite x 2 2x .Equivalent AlgebraicExpressionsTwo algebraic expressions are equivalent if, when evaluated, theyhave the same value for all replacements of the variables.Suppose that two algebraic expressions are equivalent, like the two mentioned above, x 2 and 2 x .What it means, according to the definition, is that if we choose any value of x , let‟s say x 1, andevaluate x 2 12 2, and then evaluate 2 x 21 2, the results must be the same. If we changethe value of x , for example to 4, again two results are equal ( x 2 42 8, and 2 x 24 8 ). Nomatter what the value of x , the two results are always going to be equal. Thus, to determine that twoexpressions are equivalent one would have to evaluate them for all possible sets of values of variables.Since we cannot check all, we cannot prove equivalence by performing evaluation (make sure that youunderstand that even if we determine that two expressions assume the same value for many sets ofvalues of variables, we still cannot claim that the two expressions are equivalent). To prove theequivalence of algebraic expressions, some general rules must be employed.Terms and factorsIn arithmetic, we often refer to numbers that are being added as terms, and to numbers multiplied asfactors. For example, 3 and 4 are terms of addition 3 4 7 , while 3 and 5 are called factors of 15,since35 15. We will now generalize the notions of terms and factors.TermsAlgebraic expressions that are added (or subtracted) arecalled terms. Each sign, + or –, is a part of the term that followsthe sign.In other words, the addition and subtraction signs break the expression into smaller parts, called terms,and so, in 3 x 2xy y there are three terms: 3 x , 2 xy , y . Notice that because y is preceded by a5a5aminus sign, the minus sign is a part of the term: y . The expressions a and are terms in a .d dSome expressions have just one term. For example, both 3 xy and x 3 y have only one term.24