GUIDED NOTES. College. Algebra. + Integrated. Review

GUIDED NOTES College Algebra + Integrated Review

Editor: Kara Roche Content Contributors: Daniel Breuer, Jennifer Comer Lead Designer: Tee Jay Zajac Designers: B. Syam Prasad, Patrick Thompson, James Smalls Lead Developer: Doug Chappell VP Research & Development: Marcel Prevuznak VP of Marketing: Emily Cook Cover Design: Patrick Thompson A division of Quant Systems, Inc. 546 Long Point Road, Mount Pleasant, SC 29464 Copyright 2017 by Hawkes Learning / Quant Systems, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written consent of the publisher. Printed in the United States of America ISBN: 978-1-944894-95-5

Chapter 1.R TABLE OF CONTENTS Number Systems and Fundamental Concepts of Algebra Integrated Review 1.R.1 Exponents, Prime Numbers, and LCM.................................... 2 1.R.2 Reducing Fractions.................................................... 4 1.R.3 Decimals and Percents................................................. 5 1.R.4 Simplifying Radicals.................................................... 8 Chapter 2.R Equations and Inequalities of One Variable Integrated Review 2.R.1 Multiplication and Division with Fractions................................ 10 2.R.2 Addition and Subtraction with Fractions................................. 11 2.R.3 Applications: Number Problems and Consecutive Integers................. 13 2.R.4 Solving Equations: Ratios and Proportions............................... 14 Chapter 4.R Relations, Functions, and Their Graphs Integrated Review 4.R.1 Order of Operations.................................................. 16 4.R.2 Variables and Algebraic Expressions.................................... 17 4.R.3 Simplifying Expressions............................................... 18 4.R.4 Translating Phrases into Algebraic Expressions........................... 19 Chapter 5.R Polynomial Functions Integrated Review 5.R.1 Greatest Common Factor of Two or More Terms.......................... 22 5.R.2 Factoring Trinomials by Grouping....................................... 23 5.R.3 Additional Factoring Practice........................................... 25

Chapter 6.R Rational Functions and Conic Sections Integrated Review 6.R.1 Defining Rational Expressions.......................................... 28 6.R.2 Special Products..................................................... 30 6.R.3 Special Factorizations - Squares........................................ 32 Chapter 7.R Exponential and Logarithmic Functions Integrated Review 7.R.1 Simplifying Integer Exponents I......................................... 34 7.R.2 Simplifying Integer Exponents II........................................ 36 7.R.3 Rational Exponents................................................... 38 Chapter 8.R Systems of Equations Integrated Review 8.R.1 Solving Systems of Linear Equations by Graphing......................... 42 8.R.2 Systems of Linear Inequalities.......................................... 44

Chapter 1.R Number Systems and Fundamental Concepts of Algebra Integrated Review

2 1.R.1 Exponents, Prime Numbers, and LCM 1.R.1 Exponents, Prime Numbers, and LCM Exponents Example 2: Translating Expressions with Exponents a. b. Prime Numbers and Composite Numbers DEFINITION: Prime Numbers HAWKES LEARNING. All Rights Reserved.

1.R.1 Exponents, Prime Numbers, and LCM 3 DEFINITION: Composite Numbers Prime Factorization of Composite Numbers PROCEDURE: To Find the Prime Factorization of a Composite Number 1. 2. 3. Least Common Multiple (LCM) Tests for Divisibility

4 1.R.2 Reducing Fractions 1.R.2 Reducing Fractions Rational Numbers Defined DEFINITION: Rational Number Placement of Negative Signs in Fractions Reducing Fractions To Build a Fraction to Higher Terms To Reduce a Fraction Example 4: Reducing Fractions HAWKES LEARNING. All Rights Reserved.

1.R.3 Decimals and Percents 5 1.R.3 Decimals and Percents Addition with Decimal Numbers To Add Decimal Numbers 1. 2. 3. 4. Multiplication with Decimal Numbers To Multiply Decimal Numbers 1. 2. 3.

6 1.R.3 Decimals and Percents Division with Decimal Numbers To Divide Decimal Numbers 1. 2. 3. 4. Rounding to the Right of the Decimal Point 1. 2. 3. Understanding Percent HAWKES LEARNING. All Rights Reserved.

1.R.3 Decimals and Percents 7 Decimals and Percent To Change a Decimal to a Percent Step 1: Step 2: To Change a Percent to a Decimal Step 1: Step 2: Fractions and Percents To Change a Fraction to a Percent Step 1: Step 2:

8 1.R.4 Simplifying Radicals 1.R.4 Simplifying Radicals Algebraic Expressions with Square Roots Properties of Square Roots 1. 2. Simplest Form for Square Roots Radical Expressions with Variables Square Root of x 2 Algebraic Expressions with Cube Roots Simplest Form for Cube Roots HAWKES LEARNING. All Rights Reserved.

Chapter 2.R Equations and Inequalities of One Variable Integrated Review

10 2.R.1 Multiplication and Division with Fractions 2.R.1 Multiplication and Division with Fractions Multiplication Multiplication with Fractions Division Reciprocal Division with Fractions Example 4: Dividing Fractions with Negative Numbers Notes: HAWKES LEARNING. All Rights Reserved.

2.R.2 Addition and Subtraction with Fractions 11 2.R.2 Addition and Subtraction with Fractions Addition with Like Denominators PROCEDURE: Adding Fractions with Like Denominators Addition with Unlike Denominators Adding Fractions with Unlike Denominators 1. 2. 3. Subtraction Subtracting Fractions with Like Denominators

12 2.R.2 Addition and Subtraction with Fractions Subtracting Fractions with Unlike Denominators Subtracting Fractions with Unlike Denominators 1. 2. 3. HAWKES LEARNING. All Rights Reserved.

2.R.3 Applications: Number Problems and Consecutive Integers 13 2.R.3 Applications: Number Problems and Consecutive Integers Basic Steps for Solving Applications 1. 2. 3. 4. Consecutive Integers Consecutive Integers Consecutive Even Integers Consecutive Odd Integers

14 2.R.4 Solving Equations: Ratios and Proportions 2.R.4 Solving Equations: Ratios and Proportions Proportions Proportions and Similar Triangles 1. 2. HAWKES LEARNING. All Rights Reserved.

Chapter 4.R Relations, Functions, and Their Graphs Integrated Review

16 4.R.1 Order of Operations 4.R.1 Order of Operations Rules for Order of Operations 1. 2. 3. 4. 1. 2. 3. 4. 5. 6. HAWKES LEARNING. All Rights Reserved.

4.R.2 Variables and Algebraic Expressions 17 4.R.2 Variables and Algebraic Expressions Identifying Like Terms Notes DEFINITION: Like Terms

18 4.R.3 Simplifying Expressions 4.R.3 Simplifying Expressions Combining Like Terms HAWKES LEARNING. All Rights Reserved.

4.R.4 Translating Phrases into Algebraic Expressions 19 4.R.4 Translating Phrases into Algebraic Expressions Translating English Phrases into Algebraic Expressions Key Words To Look For When Translating Phrases Addition Subtraction Multiplication Division Exponent (Powers) add subtract (from) multiply divide square of sum difference product quotient cube of plus minus times more than less than twice increased by decreased by of (with fractions and percent) less Notes

20 4.R.4 Translating Phrases into Algebraic Expressions HAWKES LEARNING. All Rights Reserved.

Chapter 5.R Polynomial Functions Integrated Review

22 5.R.1 Greatest Common Factor of Two or More Terms 5.R.1 Greatest Common Factor of Two or More Terms Greatest Common Factor of a Set of Terms Procedure for Finding the GCF of a Set of Terms 1. 2. 3. 4. HAWKES LEARNING. All Rights Reserved.

5.R.2 Factoring Trinomials by Grouping 23 5.R.2 Factoring Trinomials by Grouping The ac-method of Factoring PROCEDURE: Analysis of Factoring by the ac-method General Method Example Step 1: Multiply. Multiply. Step 2: Find two integers whose product is and whose sum is b. If this is not possible, then the trinomial is not factorable. Step 3: Rewrite the middle term using the two numbers found in Step 2 as coefficients. Find two integers whose product is and whose sum is. Step 4: Factor by grouping the first two terms and the last two terms. Step 5: Factor out the common binomial factor. This will give two binomial factors of the trinomial.

24 5.R.2 Factoring Trinomials by Grouping Tips to Keep in Mind While Factoring 1. 2. 3. 4. Notes HAWKES LEARNING. All Rights Reserved.

5.R.3 Additional Factoring Practice 25 5.R.3 Additional Factoring Practice General Guidelines for Factoring Polynomials PROCEDURE: General Guidelines for Factoring Polynomials 1. 2. a. 1. 2. 3. b. 4. 1. 2. a. b.

26 5.R.3 Additional Factoring Practice 3. a. b. c. d. c. 3. HAWKES LEARNING. All Rights Reserved.

Chapter 6.R Rational Functions and Conic Sections Integrated Review

28 6.R.1 Defining Rational Expressions 6.R.1 Defining Rational Expressions Introduction to Rational Expressions Notes Example 1: Finding Restrictions on the Variable a. Solution x Set the denominator equal to. Solve the equation. HAWKES LEARNING. All Rights Reserved.

6.R.1 Defining Rational Expressions 29 Notes: Comments about the Numerator being DEFINITION: Summary of Arithmetic Rules for Rational Numbers (or Fractions) Reducing (or Simplifying) Rational Expressions

30 6.R.2 Special Products 6.R.2 Special Products The Difference of Two Squares: DEFINITION: Difference of Two Squares DEFINITION: Perfect Squares from to Example 1: Difference of Two Squares a. difference of two squares HAWKES LEARNING. All Rights Reserved.

6.R.2 Special Products 31 Squares of Binomials: The middle term is doubled.. perfect square trinomial The middle term is doubled. perfect square trinomial

32 6.R.3 Special Factorizations - Squares 6.R.3 Special Factorizations - Squares Difference of Two Squares: Sum of Two Squares Perfect Square Trinomials: HAWKES LEARNING. All Rights Reserved.

Chapter 7.R Exponential and Logarithmic Functions Integrated Review

34 7.R.1 Simplifying Integer Exponents I 7.R.1 Simplifying Integer Exponents I The Product Rule The Product Rule for Exponents Notes: The Exponent 0 The Exponent 0 The Quotient Rule Quotient Rule for Exponents HAWKES LEARNING. All Rights Reserved.

7.R.1 Simplifying Integer Exponents I 35 Negative Exponents Rule for Negative Exponents

36 7.R.2 Simplifying Integer Exponents II 7.R.2 Simplifying Integer Exponents II Power Rule Power Rule for Exponents Rule for Power of a Product Rule for Power of a Product Rule for Power of a Quotient Rule for Power of a Quotient HAWKES LEARNING. All Rights Reserved.

7.R.2 Simplifying Integer Exponents II 37 Using Combinations of Rules for Exponents Summary of the Rules for Exponents 1. 2. 3. 4. 5. 6. 7. 8.

38 7.R.3 Rational Exponents 7.R.3 Rational Exponents n th Roots Type of Root For square roots, For cube roots, For fourth roots, Radical Notation and Exponential Notation if, then. if, then. if, then. For roots, if, then. Radical Notation Notes 1. 2. HAWKES LEARNING. All Rights Reserved.

7.R.3 Rational Exponents 39 Rational Exponents of the Form Summary of the Rules for Exponents The General Form

40 7.R.3 Rational Exponents HAWKES LEARNING. All Rights Reserved.

Chapter 8.R Systems of Equations Integrated Review

42 8.R.1 Solving Systems of Linear Equations by Graphing 8.R.1 Solving Systems of Linear Equation by Graphing Systems of Linear Equations Solving Systems of Linear Equations by Graphing 1. 2. 3. HAWKES LEARNING. All Rights Reserved.

8.R.1 Solving Systems of Linear Equations by Graphing 43 PROCEDURE: To Solve a System of Linear Equations by Graphing 1. 2. a. b. c. 3.

44 8.R.2 Systems of Linear Inequalities 8.R.2 Systems of Linear Inequalities Solving Systems of Linear Inequalities Graphically To Solve a System of Two Linear Inequalities 1. 2. 3. HAWKES LEARNING. All Rights Reserved.