Math 1010 - Lecture 4 NotesDylan ZwickFall 20091 Algebrai c ExpressionsWe introduced a lgebraic expressions at the end of our last lecture. Today,we’re going to talk about how to do some basic manipulations with theseexpressions, usually to make them simpler, and how to translate thesetypes of expressions into words and vice-versa.1.1 Combining Like TermsOne of the major ways we can simplify algebraic expressions is by com-bining like terms. Two terms in an algebraic expression are said to be liketerms if they are both constant terms or if they both have the same variablefactor. So, for example, in the algebraic expression:x2y3− 3x2y + 5x + 7x2y − 2x + 8y − 3 + 4xy + 8the terms −3x2y and 7x2y are like terms, as are the terms 5x, −2x and−3, 8. One way that we can simplify an algebraic expression is by combin-ing like terms. So, for example, we can combine (add) the terms −3x2y and7x2y to get the term 4x2y. Doing this we ’d simplify our above p olynomialto:x2y3+ 4x2y + 5x − 2x + 8y − 3 + 4xy + 8.1Doing the same thing with our other like terms we could further sim-plify the polynomial to obtain:x2y3+ 4x2y + 4xy + 3x + 8y + 5.Note that all of these polynomials are the same, in the same way that12, 5 + 7, and 1 + 2 + 3 + 6 are the same.Examples:1. Simplify the expression 18z + 14z:2. Simplify the expression −2a + 4b − 7a − b:3. Simplify the expression 3a − 5ab + 9a2+ 4ab − a:1.2 Removing Symbols of GroupingAnother way that we can sometimes simplify an expression is by remov-ing symbols of grouping. We remove the innermost groupings first, com-bine like terms, an d then continue moving out until there are no moregroupings left.A set of parentheses preceded by a minus sign can be removed if wechange the sign of all the terms in the parentheses. For example:x − (2x + 5) = x − 2x − 5 = −x − 5.If there’s a plus sign we can remove the parentheses without modifica-tion. For example:x + (2x + 5) = x + 2x + 5 = 3x + 5.We can also occasionally simplify an expression by a p p lying the dis-tributive law and then grouping like terms.21. Use the Distributive Property to simplify the expression x(5x + 2):2. Use the Distributive Property to simplify the expression y(−y + 10):3. Use the Distributive Property to simplify the expression −6x(9x−4):1.3 An Aside on SimplificationNow, it should be noted that simplifying an expression is in many waysmore of an aesthetic, or at least practical, consideration, as composed toa formal mathematical one. It’s based upon the form of an equation thatis easiest for a human to understand and manipulate, and this is a fairlysubjective criteria.For example, there are those who would say1√2+1√3is not simplified, but3√2 + 2√36is. Why the second is more, and not less, simple than the first is beyondme. Similarly, it’s more of a personal judgement which of the followingequivalent algebraic expressions:(x + 1)2andx2+ 2x + 1is simpler. Now, this isn’t to say that simplifying numbers and alge-braic expressions is not useful. It’s incredibly useful and necessary to un-derstand. But it’s useful in that it makes future computations much, mucheasier, and the human criteria of what forms are easiest to work with is themotivation driving the concept of simplification.31.4 Evaluating Algebraic ExpressionsWhen we evaluate an algebraic expression, we substitute numerical valuesfor each of the variables in the expression. This gives us a number. Youmust ma ke sure you do the same substitution for each of the occurrencesof a variable.For example, if we wanted to evaluate the algebraic expressionx2− 3x + 7at x = 2 we’d plug in 2 for x to get22− 3(2) + 7 = 5.Examples:1. Evaluate y2− y + 5 at y = 2:2. Evaluate 5 −3xat x = −6:3. Evaluate x2− xy + y2at x = 2, y = −1:2 Constructing Algebraic ExpressionsThe idea behind constructing an algebraic expression is that we take averbal expression or relation, and translate it into an algebraic expression.While this may see m a bit obvious, the idea that you can do this was inmany ways the first big idea in algebra, and it took mankind a long timeto figure it out!42.1 Translating PhrasesThe idea behind translating a phrase is to take a verbal description of amathematical equation or relationship, and find the correct correspondingequation or relationship. We have to keep in mind when we do this thatmany words are synonyms for the same relation or operation. For exam-ple, saying the product of a and b is the same as saying a multiplied byb.Examples:1. Write “twelve more than a number n” as an a lgebraic expression:2. Write “seven-fifths of a n umbe r n” a s an algebraic expression:3. Write “the product of a number y and 10 is decreased by 35” as analgebraic expression:Now, we’re not always given a verbal d escription of the name of thevariable. Sometimes, we have to come up with the variable name our-selves. For example, if we’re asked to write “the sum of 7 and a num-ber” we’re not told what variable name we should use for the unknownnumber. In this situation, we just pick a variable name. So, we’d say ourvariable name is x, and write 7 + x.1. Write “the sum of 7 and twice a number, all divided by 8” as analgebraic expression:2. Write “the absolute value of a quotient of a number and 4” as analgebraic expression:Now, as you might imagine, this can go the other way as well. Thatis, you can be given an algebraic expression and be asked to write it as averbal phrase. As mentioned earlier, given the ambiguities of the Englishlanguage, there are frequently ma ny ways of saying the same thing, andtranslating algebraic expressions is no exception.So, for example, if we wanted to express 3n + 7 verbally we could say“the product of 3 and a number n, increased by 7.” We could also say “the5sum of 7 and 3 times a number n.” There are many other possibilities. Aslong as your verbal phrase is correct and unambiguous, it’s fine.Examples:1. Express 2 − 3x as a verbal phrase:2. Express 8(x − 5) as a verbal phrase:3. Express3 − n9as a verbal phrase:2.2 Constructing a Mathematical ModelThe ability to translate a verbal phrase into a m a thema tical model is oneof the most useful and fundamental applications of algebra. The textbooklays out a three step process for creating a mathematical model:1. Construct a verbal model that represents the problem situation.2. Assign labels to all quantities in the verbal model.3. Construct a