R.1 R.2 R.3 R.4 R.5 R.6 R.7 R.1 Sets Basic Definitions ▪ Operations on Sets Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

Classification of Numbers (N) Natural Numbers {1,2,3,4,…} (W) Whole Numbers {0,1,2,3,4,…} (Z) Integers {…-3,-2,-1,0,1,2,3,…} (Q) Rational Numbers: Numbers which can be expressed as a fraction ex: 2, 3.5, -7½, … (I) Irrational Numbers: Real numbers which can NOT be expressed as a fraction ex: √20, π, e, … (R) All of the above numbers are Real Numbers Ex. 1 Write the elements belonging to: {x|x is a natural number between 8 and 12} {9, 10, 11} Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.1 Example 2 Finding the Complement of a Set (page 4) Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A= {2, 4, 6, 8}, B = {3, 6, 9} Find A′, B′ A′ contains the elements of U that are not in A: {1, 3, 5, 7, 9} B′ contains the elements of U that are not in B: {1, 2, 4, 5, 7, 8} Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.1 Example 3 Finding the Intersection of Two Sets (page 4) 30 is the only element in both sets. Answer: = {30} (b) {3, 6, 9, 12, 15, 18} ∩ {6, 12, 18, 24} The elements 6, 12, and 18 belong to both sets. Answer: {6, 12, 18} Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.1 Example 4 Finding the Union of Two Sets (page 4) (a) {1, 3, 5, 7, 9} U {3, 6, 9, 12} This is the Union of two sets, which means the joining of the sets. = {1, 3, 5, 7, 9, 6, 12} = {1, 3, 5, 6, 7, 9, 12} (b) {10, 11, 12, 14} U {9, 10, 12, 13} = {10, 11, 12, 14, 9, 13} = {9,10, 11, 12, 13, 14} Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.1 Summary Classification of numbers Real Rational or Irrational Integers Whole Natural Sets Complement Union Intersection Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

Real Numbers and Their Properties Sets of Numbers and the Number Line ▪ Exponents ▪ Order of Operations ▪ Properties of Real Numbers ▪ Order on the Number Line ▪ Absolute Value Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

Lesson R-2 Properties of Real Numbers Rational Numbers Integers Whole Numbers Natural Numbers Irrational Numbers R #’s which can be written as a fraction Q I #’s which can NOT be written as a fraction. ex √5, π, e, 2√3, 3.121121112…, Z = { …-2,-1,0,1,2…} W = {0,1,2,3,…} N = {1,2,3,4,…} Symbols 1st

List the elements of S that belong to each set. R.2 Example 1 Identifying Elements of Subsets of the Real Numbers (page 8) List the elements of S that belong to each set. (a) natural numbers (b) whole numbers (c) integers Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

List the elements of S that belong to each set. R.2 Example 1 Identifying Elements of Subsets of the Real Numbers (cont.) List the elements of S that belong to each set. (d) rational numbers All elements of S except (e) irrational numbers (f) real numbers All elements of S are real numbers. Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

Evaluate each expression and identify the base and the exponent. R.2 Example 2 Evaluating Exponential Expressions (cont.) Evaluate each expression and identify the base and the exponent. (a) Base: 3 Exponent: 4 (b) Base: 5 Exponent: 2 (c) Base: 10 Exponent: 2 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

Evaluate R.2 Example 3(a,b) Using Order of Operations (page 11) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

Evaluate R.2 Example 3(c,d) Using Order of Operations (cont.) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

Evaluate using a = –4, b = 3, and c = –6. R.2 Example 4(a) Using Order of Operations (page 11) Evaluate using a = –4, b = 3, and c = –6. Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

Evaluate using a = –4, b = 3, and c = –6. R.2 Example 4(b) Using Order of Operations (cont.) Evaluate using a = –4, b = 3, and c = –6. Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.2 Example 5(a) Using the Commutative and Associative Properties to Simplify Expressions (page 13) a) Simplify (12 + 2x) + 18 = 2x + 30 b) Simplify = -20t c) Simplify = 24s Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.2 Example 6 Using the Distributive Property (page 14) Rewrite using the distributive property and simplify. (a) (b) = 3r – 5s (c) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.2 Example 7 Evaluating Absolute Values (page 15) Evaluate each expression: (a) (b) (c) (d) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.2 Example 9 Evaluating Absolute Value Expressions (page 16) Let m = 13 and n = –9. Evaluate each expression. (a) – (b) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.2 Summary Classification of Numbers: Real (R) Rational (Q) or Irrational (I) Integers (Z) Whole (W) Natural (N) Order of Operations: PEMDAS Simplify Expression: Distribute and combine like terms Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.3 Polynomials Rules for Exponents ▪ Polynomials ▪ Addition and Subtraction ▪ Multiplication ▪ Division Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

Find each product. (a) (b) R.3 Example 1 Using the Product Rule (page 22) Find each product. (a) (b) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

Simplify. Assume all variables represent nonzero real numbers. R.3 Example 2 Using the Power Rules (page 22) Simplify. Assume all variables represent nonzero real numbers. (a) (b) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

Simplify. Assume all variables represent nonzero real numbers. R.3 Example 2 Using the Power Rules (cont.) Simplify. Assume all variables represent nonzero real numbers. (c) (d) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.3 Example 3 Using the Definition of a0 (page 24) Evaluate each power. (a) 80 = 1 33 = 27 32 = 9 31 = 3 30 = 3-1 = (b) –80 = -1 1 (c) (–8)0 = 1 1/3 (d) –(– 8)0 = -1 (e) (–3b8)0 = 1 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.3 Example 4 Adding and Subtracting Polynomials (page 24) Add or subtract. (a) (b) = (c) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.3 Example 4 Adding and Subtracting Polynomials (cont.) Add or subtract. (d) Multiply (e) The company has a mistake here and we will cover multiplication in the next slides Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.3 Example 6(a) Using FOIL to Multiply Two Binomials (page 27) Find the product. 7y + 3 4y –5 28y2 +12y -35y -15 = 28y2 – 35y + 12y – 15 6p + 11 6p –11 36p2 +66p = 36p2 – 66p + 66p – 121 -66p -121 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.3 Example 6(c) Using FOIL to Multiply Two Binomials (cont.) Find the product. 2x – 5 2x +5 4x2 -10x +10x -25 = x3 (4x2 – 25) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.3 Example 7 Using the Special Products (page 27) Find each product. (a) (b) (c) These are all examples of the product of the sum and difference of two terms. Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.3 Example 7 Using the Special Products (cont.) 8z + 3 Find each product. 8z +3 64z2 +24z +24z +9 (d) = (8z + 3) (8z + 3) = 64z2 + 24z + 24z + 9 These are examples of the square of a binomial. (e) = (5z – 12q3)(5z – 12q3) = 25z2 – 60zq3 – 60zq3 – 144q6 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.3 Example 8(a) Multiplying More Complicated Binomials (page 28) Find the product: = (4x – 3) (4x – 3) – 49y2 = 16x2 – 12x – 12x + 9 – 49y2 = 16x2 – 24x + 9 – 49y2 This is an more complicated example of the product of the sum and difference of two terms. Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.3 Example 8(b) Multiplying More Complicated Binomials (cont.) Find the product: a2 -2ab +b2 a2 -2ab +b2 a4 -2a3b +a2b2 -2a3b +4a2b2 -2ab3 +a2b2 -2ab3 +b4 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.3 Example 8(c) Multiplying More Complicated Binomials (cont.) Find the product: s2 +8st +16t2 s +4t s3 +8s2t +16st2 +4s2t +32st2 +64t3 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.3 Example 9 Dividing Polynomials (page 29) Divide 4n2 + n + 1 –(12n3 + 8n2) 12n3 + 8n2 3n2 + 5n – 8 –(3n2 + 2n) 3n2 + 2n 3n – 8 –(3n + 2) 3n + 2 –10 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.3 Example 10 Dividing Polynomials with Missing Terms (page 29) Divide Insert placeholders for missing terms They do not put the minus sign… I prefer to. Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.3 .. Add to notebook Add to notebook Simplify: When dividing by a monomial separate each term of the numerator over the denominator then reduce Add to notebook Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.3 Summary Monomials: multiplication, division, and raise to a power Polynomials: addition, subtraction, multiplication, and division Division by a binomial use long division Division by a monomial separate each numerator over the denominator and then reduce Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

Factoring Polynomials Factoring Out the Greatest Common Factor ▪ Factoring by Grouping ▪ Factoring Trinomials ▪ Factoring Binomials ▪ Factoring by Substitution Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.4 Example 1 Factoring Out the Greatest Common Factor (page 34) Factor out the greatest common factor from each polynomial. (a) (b) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.4 Example 1 Factoring Out the Greatest Common Factor (cont.) Factor out the greatest common factor from the polynomial. (c) GCF = 2(x – 2) = 2(x – 2) [ 12(x – 2)2 – 8(x – 2) + 3] Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.4 Example 2(a) Factoring By Grouping (page 35) Factor by grouping. Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.4 Example 3(a) Factoring Trinomials (page 36) Factor , if possible. Multiply first and last = -60 Look at factors of -60 with difference of middle term of + 4 -60 1*60 2*30 3*20 4*15 5*12 6*10 (-6)(10) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.4 Example 3(b) Factoring Trinomials (page 36) -36 1*36 2*18 3*12 4*9 6*6 Factor , if possible. F x L = -36 with diff of -5 = 4t(3t + 1) – 3(3t + 1) = (4t – 3)(3t + 1) (4)(-9) 48 1*48 2*24 3*16 4*12 6*8 Factor , if possible. F x L = 48 with sum of -15 Not possible, so 3x² - 15x + 16 is prime Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.4 Example 3(d) Factoring Trinomials (page 36) Factor , if possible. Factor out the GCF, 3, first: FxL = 40 sum of 14 40 1*40 2*20 4*10 5*8 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.4 Example 4 Factoring Perfect Square Trinomials (page 37) Factor each: (a) (b) To Check for Perfect Square Trinomials: Take the square root of the first term Take the square root of the last term Multiply the two square roots and double the product If the product equals the middle term then it is a perfect square Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.4 Example 5 Factoring Differences of Squares (page 38) Factor each: (a) (b) (c) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.4 Example 5 Factoring Differences of Squares (cont.) Factor each: (d) (e) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.4 Example 6 Factoring Sums or Differences of Cubes (page 39) Factoring sum/diff of cubes A3 – B3 = (A – B)(A² + AB + B²) A3 + B3 = (A + B)(A² – AB + B²) Factor each polynomial: (a) = (r – 2s)(r2 + 2rs + 4s2) (b) (c)

R.4 Example 7(a) Factoring by Substitution (page 40) FxL= -90, diff of -1: 1*90 2*45 3*30 5*18 6*15 9*10 Replace m2 with u: =5u(3u – 2) +3(3u – 2) = (5u + 3)(3u – 2) Replace u with m2: = (5m2 + 3)(3m2 – 2) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.4 Example 7(b) Factoring by Substitution (page 40) Replace (3x + 1) with u: FxL = -200, diff of 10: Replace u with (3x + 1), then simplify: = (4(3x + 1) – 5) (2(3x + 1) + 5) = (12x + 4 – 5) (6x + 2 + 5) = (12x – 1)(6x + 7) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.4 Example 7(c) Factoring by Substitution (page 40) Replace (3x + 1) with u: Replace u with (3x + 1), then simplify: Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.4 Summary Factoring: always check for GCF before doing anything Special Binomials: Difference of Squares: a2 – b2 = (a – b)(a + b) Difference of Cubes: a3 – b3 = (a – b)(a2 + ab + b2) Sum of Cubes : a3 + b3 = (a + b)(a2 – ab + b2) Special Trinomials: Perfect Square: a2 + 2ab + b2 = (a + b)2 a2 – 2ab + b2 = (a – b)2 Trinomials: Multiple F•L, find sum/difference of middle terms and then take out GCF’s Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.5 Rational Expressions Rational Expressions ▪ Lowest Terms of a Rational Expression ▪ Multiplication and Division ▪ Addition and Subtraction ▪ Complex Fractions Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

Write the rational expression in lowest terms. R.5 Example 1(a) Writing Rational Expressions in Lowest Terms (page 44) Write the rational expression in lowest terms. (a) (b) -1 1 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.5 Example 2(a) Multiplying or Dividing Rational Expressions (page 45) z4 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.5 Example 2(c) Multiplying or Dividing Rational Expressions (page 45) Divide. Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.5 Example 2(d) Multiplying or Dividing Rational Expressions (page 45) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.5 Example 3(a) Adding or Subtracting Rational Expressions (page 47) Find the LCD: Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.5 Example 3(b) Adding or Subtracting Rational Expressions (page 47) Find the LCD: Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.5 Example 3(c) Adding or Subtracting Rational Expressions (page 47) Find the LCD: Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.5 Example 4(a) Simplifying Complex Fractions (page 49) Multiply the numerator and denominator by the LCD of all the fractions, x2. Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.5 Example 4(b) Simplifying Complex Fractions (page 49) Multiply the numerator and denominator by the LCD of all the fractions, z(z + 1)(z – 1). Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.5 Summary Simplify Rational Expressions: Factor then reduce common factors Multiply and Divide Rational Expressions: Watch out for opposite factors such as (x – 3) and (3 – x) Add and Subtract Rational Expressions: Get a LCD, then add numerators Simplify Complex Rational Expressions: Multiply numerator and denominator by the LCD of all fractions within the problem and then simplify Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.6 Rational Exponents Negative Exponents and the Quotient Rule ▪ Rational Exponents ▪ Complex Fractions Revisited Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.6 Example 1 Using the Definition of a Negative Exponent (page 53) Evaluate each expression. (d) (a) (b) (e) (c) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.6 Example 2 Using the Quotient Rule (page 54) Simplify each expression. (a) (b) = y4y9 (c) (d) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.6 Example 3(a&b) Using Rules for Exponents (page 54) Simplify. Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.6 Example 3(c) Using Rules for Exponents (page 54) Simplify. Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.6 Example 4 Using the Definition of a1/n (page 55) Evaluate each expression. (a) (b) (c) (d) (e) (f) not a real number (g) (h) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.6 Example 5 Using the Definition of am/n (page 56) Evaluate each expression. (a) (b) (c) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.6 Example 5 Using the Definition of am/n (cont.) Evaluate each expression. (d) (e) is not a real number because is not a real number. (f) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

=18 Simplify each expression. (a) (b) (c) R.6 Example 6 Combining the Definitions and Rules for Exponents (page 57) Simplify each expression. (a) =18 (b) (c) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

Simplify each expression. R.6 Example 6 Combining the Definitions and Rules for Exponents (cont.) Simplify each expression. (d) (e) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

Factor out the least power of the variable or variable expression. R.6 Example 7 Factoring Expressions with Negative or Rational Exponents (page 58) Factor out the least power of the variable or variable expression. (a) (b) (c) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.6 Example 8 Simplifying a Fraction with Negative Exponents (page 58) Simplify. Write the result with only positive exponents. xy Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.6 Summary Simplify fractions with negative exponents Raise numbers to the m/n power Multiply, divide and raise monomials to powers Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.7 Radical Expressions Radical Notations ▪ Simplified Radicals ▪ Operations with Radicals ▪ Rationalizing Denominators Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.7 Example 1 Evaluating Roots (page 63) Write each root using exponents and evaluate. (a) (b) (c) (d) is not a real number. Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.7 Example 1 Evaluating Roots (cont.) Write each root using exponents and evaluate. (e) (f) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.7 Example 2 Converting From Rational Exponents to Radicals (page 63) Write in radical form and simplify. (a) (b) (c) (d) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.7 Example 2 Converting From Rational Exponents to Radicals (cont.) Write in radical form and simplify. (e) (f) (g) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.7 Example 3 Converting From Radicals to Rational Exponents (page 63) Write in exponential form. (a) (b) (c) (d) (e) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.7 Example 4 Using Absolute Value to Simplify Roots (page 64) Simplify each expression. Even, Even, Odd get │ │ (a) (b) (c) (d) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.7 Example 4 Using Absolute Value to Simplify Roots (cont.) Simplify each expression. (e) (f) (g) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

Simplify each expression. R.7 Example 5 Using the Rules for Radicals to Simplify Radical Expressions (page 65) Simplify each expression. (a) (b) (c) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

Simplify each expression. R.7 Example 5 Using the Rules for Radicals to Simplify Radical Expressions (cont.) Simplify each expression. (d) (e) (f) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.7 Example 6 Simplifying Radicals (page 66) Simplify each radical. (a) (b) (c) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.7 Example 7 Adding and Subtracting Like Radicals (page 66) Add or subtract as indicated. (a) (b) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.7 Example 7 Adding and Subtracting Like Radicals (cont.) Add or subtract as indicated. (c) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

Simplify each radical. (a) (b) (c) R.7 Example 8 Simplifying Radicals by Writing Them With Rational Exponents (page 67) Simplify each radical. (a) (b) (c) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.7 Example 9(a) Multiplying Radical Expressions (page 68) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.7 Example 10 Rationalizing Denominators (page 68) Rationalize each denominator. (a) (b) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

Simplify the expression. R.7 Example 11(a) Simplifying Radical Expressions with Fractions (page 69) Simplify the expression. Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

Simplify the expression. R.7 Example 11(b) Simplifying Radical Expressions with Fractions (page 69) Simplify the expression. Quotient rule Simplify the denominators. Write with a common denominator. Subtract the numerators. Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.7 Example 12 Rationalizing a Binomial Denominator (page 69) Rationalize the denominator. Multiply the numerator and denominator by the conjugate of the denominator. Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

R.7 Summary Nth root Raise to the m/n power Rewrite from radical from to rational exponents Simplify radical expressions When to use absolute value bars when simplifying radical expressions even – even – odd use absolute value bars Add, subtract, & multiple radical expression Rationalize denominators for monomials Rationalize denominators for binomials multiple the numerator and denominator by the conjugate of the denominator Copyright © 2008 Pearson Addison-Wesley. All rights reserved.