Math Review for Algebra and Precalculus Stanley Ocken Department of Mathematics The City College of CUNY Copyright © January 2007 by Stanley Ocken No part of this document may be copied or reproduced[.]
Math Review for Algebra and Precalculus Stanley Ocken Department of Mathematics The City College of CUNY Copyright © January 2007 by Stanley Ocken No part of this document may be copied or reproduced in any form whatsoever without express permission of the author Math Review for Algebra and Precalculus Stanley Ocken Department of Mathematics The City College of CUNY Copyright © January 2007 Copyright © January 2007 by Stanley Ocken No part of this document may be copied or reproduced in any form whatsoever without express permission of the author Table of Contents Part I: Algebra Notes for Math 195 Introduction…………………………………… Basic algebra laws; order of operations………………… How algebra works…………………………….……….… 12 Simplifying polynomial expressions………………………29 Functions………………………………………………… 41 When to use parentheses………………………………… 55 Working with fractions…………………………………… 61 Adding fractions………………………………………… 76 Copyright © January 2007 by Stanley Ocken No part of this document may be copied or reproduced in any form whatsoever without express permission of the author Math Review for Precalculus and Calculus Part I: Algebra Introduction Algebra is the language of calculus, and calculus is needed for science and engineering When you attack a real-world problem, you want to represent the problem using algebra expressions When you read technical books, you want to be comfortable deciphering and working with these expressions Computers can’t either of these tasks for you Algebra used in undergraduate mathematics involves three main activities: rewriting expressions, solving equations, and solving inequalities You need to perform these somewhat mechanical activities quickly and accurately It’s very difficult to achieve this goal unless you understand how algebra works Algebra is a symbolic language that allows communication between people who don’t know each others’ spoken language The grammar of the language involves three main components: expressions, identities, and equations An expression involves numbers, variables, parentheses, and algebra operations Basic types of expressions are integers, variables, monomials, polynomials, and so forth We’ll deal mostly with expressions in one variable, such as the polynomial x − x + An identity between two expressions, written with an equals sign, is a statement that each expression can be obtained by rewriting the other A simple example is x + = + x + With rare exceptions, substituting numbers for variables turns an identity into a true statement about numbers For example, setting x to yields + = + + An important part of algebra is using identities to rewrite expressions An equation is also a statement that two expressions are equal In most equations, however, equality holds only for specific values of the variable For example, the statement x + = is true only when x is or – We say that the solutions of the equation x + = are x = and x = – Please remember;: we rewrite expressions but we solve equations In this preliminary edition, section headings such as AN1 are used for Algebra Notes, Section Copyright © January 2007 by Stanley Ocken No part of this document may be copied or reproduced in any form whatsoever without express permission of the author CHAPTER 1: BASIC ALGEBRA LAWS; ORDER OF OPERATIONS 1.1 Algebra operations and notation Let’s begin with two tricky examples Example 1.1.1: Rewrite x −1 as an expression with no negative powers ⎛1⎞ Right: x −1 = 2⎜ ⎟ = Wrong: x −1 = 2x ⎝ x⎠ x Example 1.1.2: Simplify the expression − Right: − = −(5 ) = −25 Wrong: − = ( −5) = 25 To understand what’s going on, we need to review in some detail the five algebra operations: addition, subtraction, multiplication, division, and exponentiation Each of these is called a binary operation because it is used to combine two expressions The table below lists notation and terminology for these operations The last entry shows a special operation called negation, which operates on one expression and is an abbreviation for multiplication by –1 Operation Addition Subtraction Write 4+3 4–3 Multiplication 4⋅ Traditional notation Say plus subtract or minus times Describe the answer The sum of and The difference of and The product of and divided by four thirds over (slang) The quotient of by (raised) to the 3rd (power) The 3rd power of 4 is the base is the exponent The negation (additive opposite) of is –3 4× Traditional notation * Calculator notation Division Exponentiation Negation 4(3 + x ) Implied times sign xy Implied times sign ÷ Seldom used / Calculator notation Traditional notation Traditional notation 4^3 Calculator notation –3 Negative or Minus Negative of minus – (–5) = It’s a bit annoying that the minus sign ‘–’ is used for three different purposes: naming a negative number, subtraction, and negation Specifically: Copyright © January 2007 by Stanley Ocken No part of this document may be copied or reproduced in any form whatsoever without express permission of the author x as a reduced fraction Example 7.6.2 Rewrite x− x Solution: Multiply both numerator and denominator by x x+ ( x)⎛ x + ⎞ x( x) + x⎛ ⎞ ⎟ ⎜ ⎟ ⎜ x⎠ ⎝ x ⎠ = x +1 ⎝ x = = 1⎞ ⎛ ⎞ x2 −1 ⎛ x− ( x)⎜ x − ⎟ x( x) − x⎜ ⎟ x x⎠ ⎝ x⎠ ⎝ The fraction doesn’t reduce, since x + doesn’t factor x+ x + as a reduced fraction Example 7.6.3 Rewrite 1+ x −1 1+ Solution: Multiply top and bottom by the LCD of the fractions case the LCD is the product ( x + 1)( x − 1) Then ⎛ ( x + 1)( x − 1)⎜1 + ⎝ ⎛ ( x + 1)( x − 1)⎜1 + ⎝ 1 and In this x +1 x −1 ⎞ ⎟ x + 1⎠ ⎞ ⎟ x − 1⎠ ⎛ ⎞ ( x + 1)( x − 1)(1) + ( x + 1)( x − 1)⎜ ⎟ x + 1⎠ ⎝ = ⎛ ⎞ ( x + 1)( x − 1)(1) + ( x + 1)( x − 1)⎜ ⎟ ⎝ x − 1⎠ ( x + 1)( x − 1)(1) + ( x − 1) Don' t multiply out! Factor the top and bottom to see if you can cancel ( x + 1)( x − 1)(1) + ( x + 1) = ( x − 1)[( x + 1) + 1] ( x + 1)[( x − 1) + 1] = ( x − 1)( x + 2) ( x + 1) x 90 Copyright © January 2007 by Stanley Ocken No part of this document may be copied or reproduced in any form whatsoever without express permission of the author Exercise 7.6.1 Rewrite each nested fraction as a fraction in standard form a) b) c) d) e) f) g) + x y + x y + 1 + + x+2 x−2 + x+2 x−2 + a b + c d 1 + x −1 x −1 1 + x −1 x +1 1 + x + 2x − x −9 + ab a + ba b 91 ... any form whatsoever without express permission of the author Math Review for Precalculus and Calculus Part I: Algebra Introduction Algebra is the language of calculus, and calculus is needed for. .. may be copied or reproduced in any form whatsoever without express permission of the author Math Review for Algebra and Precalculus Stanley Ocken Department of Mathematics The City College of CUNY... copied or reproduced in any form whatsoever without express permission of the author Table of Contents Part I: Algebra Notes for Math 195 Introduction…………………………………… Basic algebra laws; order of