Math 101 Prerequisite Review Basic operations and equation solving are prerequisites for this course Certain approaches for simplifying expressions and performing operations with different types of numbers are universal that is these processes will always work However alternative approaches may be more efficient in certain circumstances The art of algebra involves choosing a strategy to take advantage of simplicity Knowing different options for solution processes will help you make efficient choices The material in Chapters 1 2 of the OpenStax text and in Appendix A are provided to review these skills 1 Basics of Algebra Number Systems Classify each number OSCA Section 1 1 OSCA pg 5 N W I Q Q 36 a 8 3 73 b c d 5 e 3 2121121112 Order of Operations To evaluate mathematical expressions with mixed operations follow the standard or der of operations Follow the Order of Operations to evaluate expressions 1 Simplify any expression within grouping symbols 2 Simplify any expressions containing exponents or radicals 3 Perform any multiplication and division in order from left to right 4 Perform any addition and subtraction in order from left to right 2 Example Evaluate using the order of operations cid 18 1 4 cid 19 2 3 2 cid 19 2 3 2 cid 18 5 2 25 9 16 2 3 2 4 25 4 6 25 3 8 11 25 Practice Use the order of operations to evaluate the expression Simplify grouping symbols under the radical fraction numerator Simplify exponents radicals Division then multiplication left to right order Subtraction then addition left to right order 1 64 8 4 2 2 5 6 4 11 3 4 3 18x 9 12 4 18y 2 1 7y 5 2 16 8 4 2 2 6 cid 0 9 6t 4 cid 1 2 3 Properties of Real Numbers The commutative property for addition and multiplication states that numbers can be added multiplied in any order without affecting the result That is 2 5 5 2 and 7 4 4 7 The associative property for addition and multiplication states that it does not matter how we group num bers when adding multiplying more than 2 at a time That is 2 3 7 2 3 7 or 2 10 5 7 12 Also 4 5 6 4 5 6 or 20 6 4 30 120 The distributive property states that the product of a factor times a sum is the sum of the product of the factor times each term in the sum That is 3 10 7 3 10 3 7 30 21 51 Use the commutative associative and or distributive properties to simplify 7 Commutative Property x3 3x and 2 x x 2 8 5 8 8 Associative Property 5 8 8 cid 19 cid 19 2 cid 18 2 cid 18 4 3 7 4 7 3 9 7 4 4 7 Associative Property 10 100 0 75 2 38 Distributive Property 100 0 75 100 2 38 75 238 11 3 6 3 4 Distributive Property backwards factor 3 6 4 4 Basics 1 4 2 4 3 2x 4 4y 2 5 2 6 3t 8 7 see problem 8 5 0 5 9 1 2 2 3 3 10 163 11 3 10 30 Answers for Practice Exercises 5 Polynomial Operations Add Subtract Multiply Add or Subtract polynomial expressions by adding or subtracting the coefficients of the terms with the same variable components Combine Like Terms OSCA Sections 1 4 1 5 4x 7x 3y 11x 3y p2 5p 3p2 5p 2p2 3wq 7w 2q 3wq 7w 2q no like terms Multiply polynomials Patterns Distribute multiplication over addition Then combine like terms binomial binomial FOIL ax b cx d acx2 adx bcx bd combine like terms perfect square binomials x a 2 x2 2ax a2 difference of 2 squares a b a b a2 b2 Practice 1 12x2 3x 8x2 19x 2 6t2 14t 7 2t2 t 11 3 Subtract 4w2 13w from 2w3 w2 4 4 2y 3 4y 5 5 3y c 3y c 6 3x2 2x 1 2x 7 7 x 2 3 6 Factoring OSCA Section 1 5 Factor out the GCF Greated Common Factor first Use the distributive property backwards Binomial patterns a2 b2 a b a b a2 b2 prime a3 b3 a b a2 ab b2 a3 b3 a b a2 ab b2 Factorable trinomials More than 3 terms Try factoring by grouping Practice Factor the expressions completely 8 y2 20y 100 9 121w2 169 10 6p4 2p3 3p2 p 11 2a2 9a 18 12 18xy2 24xy 10x 7 Rational Expressions OSCA Section 1 6 To add or subtract rational expressions transform each expression so that there is a common denominator Then add or subtract the numerators Keep in mind Transforming to a common denominator means multiplying by 1 The value of the expression does The fraction bar is a grouping symbol Use parentheses to avoid sign errors when combining the 21 35 10 35 21 10 35 31 35 not change numerators Practice 13 3 5 2 7 cid 18 3 cid 19 5 cid 18 2 cid 19 7 cid 19 7 cid 18 7 cid 18 3y cid 19 5 cid 19 cid 18 5 cid 19 cid 18 2 2 14 3 2y 5 3y2 3 2y 3 3y 6y2 3y 5 2 6y2 5 3y2 15 1 x 3 4 x 5 1 x 3 4 x 5 x 3 x 5 x 3 x 5 cid 18 x 5 cid 19 x 5 cid 19 cid 18 x 3 x 3 16 p 2 3 p 4 4 17 4 a 1 5 a 1 8 To multiply or divide rational expressions factor all numerators and denominators first Then divide out the common factors a factor divided by itself 1 Division with rational expressions is equivalent to mul tiplying by the reciprocal of the divisor Practice 18 x2 16 x2 5x 4 19 2x2 7x 4 2x2 5x 12 20 x2 x 2 x 3 x2 9 x 1 9 21 4 x2 10 x3 4 x2 x3 10 22 y2 9 3y 6 6y 18 y2 4 23 q 6 3 p 2q 3p 24 x y 1 1 x y 10 Polynomial Operations Answers for Practice Exercises 1 20x2 16x 2 8t2 13t 18 3 2w3 5w2 13w 4 4 8y2 2y 15 5 9y2 c2 6 6x3 17x2 16x 7 7 x3 6x2 12x 8 8 y 10 2 9 11w 13 11w 13 10 3p 1 2p2 1 p 11 2a 3 a 6 12 2x 3y 5 3y 1 13 see problem 14 15 16 17 18 9y 10 6y2 3x 17 x 3 x 5 p 20 12 9a 1 a2 1 x 4 x 1 2x 1 2x 3 19 20 x 2 x 3 21 2x 5 22 y 3 y 2 18 18 qp 4q 23 24 xy 11 Vocabulary equivalent equations conditional contradiction identity absolute value Solving Equations Objectives 1 Solve linear equations in one variable analytically 2 Solve equations that lead to linear equations 3 Check solutions to eliminate invalid values 4 Solve …