Algebra II For Dummies

by Sterling, Mary Jane

• ISBN13: 9780471775812
• ISBN10: 0471775819
• Format: Paperback
• Copyright: 2006-06-19
• Publisher: For Dummies

Summary

Besides being an important area of math for everyday use, algebra is a passport to studying subjects like calculus, trigonometry, number theory, and geometry, just to name a few. To understand algebra is to possess the power to grow your skills and knowledge so you can ace your courses and possibly pursue further study in math. Algebra II For Dummies is the fun and easy way to get a handle on this subject and solve even the trickiest algebra problems. This friendly guide shows you how to get up to speed on exponential functions, laws of logarithms, conic sections, matrices, and other advanced algebra concepts. In no time you'll have the tools you need to: Interpret quadratic functions Find the roots of a polynomial Reason with rational functions Expose exponential and logarithmic functions Cut up conic sections Solve linear and non linear systems of equations Equate inequalities Simplifyy complex numbers Make moves with matrices Sort out sequences and sets This straightforward guide offers plenty of multiplication tricks that only math teachers know. It also profiles special types of numbers, making it easy for you to categorize them and solve any problems without breaking a sweat. When it comes to understanding and working out algebraic equations, Algebra II For Dummies is all you need to succeed!

Author Biography

Mary Jane Sterling has authored Algebra For Dummies, Trigonometry For Dummies, Algebra Workbook For Dummies, Trigonometry Workbook For Dummies, Algebra I CliffsStudySolver, and Algebra II CliffsStudySolver. She taught junior high and high school math for many years before beginning her current 25-year-and-counting career at Bradley University in Peoria, Illinois. Mary Jane enjoys working with her students both in the classroom and outside the classroom, where they do various community service projects.

Table of Contents

Introduction	1	(1)
About This Book	1	(1)
Conventions Used in This Book	2	(1)
Foolish Assumptions	2	(1)
How This Book Is Organized	3	(2)
Part I: Homing in on Basic Solutions	3	(1)
Part II: Facing Off with Functions	4	(1)
Part III: Conquering Conics and Systems of Equations	4	(1)
Part IV: Shifting into High Gear with Advanced Concepts	5	(1)
Part V: The Part of Tens	5	(1)
Icons Used in This Book	5	(1)
Where to Go from Here	6	(1)
Part I: Homing in on Basic Solutions	7	(90)
Going Beyond Beginning Algebra	9	(14)
Outlining Algebra Properties	10	(3)
Keeping order with the commutative property	10	(1)
Maintaining group harmony with the associative property	10	(1)
Distributing a wealth of values	11	(1)
Checking out an algebraic ID	12	(1)
Singing along in-verses	13	(1)
Ordering Your Operations	13	(1)
Equipping Yourself with the Multiplication Property of Zero	14	(1)
Expounding on Exponential Rules	15	(2)
Multiplying and dividing exponents	15	(1)
Getting to the roots of exponents	15	(1)
Raising or lowering the roof with exponents	16	(1)
Making nice with negative exponents	17	(1)
Implementing Factoring Techniques	17	(6)
Factoring two terms	17	(1)
Taking on three terms	18	(4)
Factoring four or more terms by grouping	22	(1)
Toeing the Straight Line: Linear Equations	23	(14)
Linear Equations: Handling the First Degree	23	(5)
Tackling basic linear equations	24	(1)
Clearing out fractions	25	(1)
Isolating different unknowns	26	(2)
Linear Inequalities: Algebraic Relationship Therapy	28	(4)
Solving basic inequalities	28	(1)
Introducing interval notation	29	(1)
Compounding inequality issues	30	(2)
Absolute Value: Keeping Everything in Line	32	(5)
Solving absolute-value equations	32	(2)
Seeing through absolute-value inequality	34	(3)
Cracking Quadratic Equations	37	(20)
Solving Simple Quadratics with the Square Root Rule	38	(1)
Finding simple square-root solutions	38	(1)
Dealing with radical square-root solutions	38	(1)
Dismantling Quadratic Equations into Factors	39	(4)
Factoring binomials	39	(2)
Factoring trinomials	41	(1)
Factoring by grouping	42	(1)
Resorting to the Quadratic Formula	43	(3)
Finding rational solutions	44	(1)
Straightening out irrational solutions	44	(1)
Formulating huge quadratic results	45	(1)
Completing the Square: Warming Up for Conies	46	(3)
Squaring up to solve a quadratic equation	46	(2)
Completing the square twice over	48	(1)
Getting Promoted to High-Powered Quadratics (without the Raise)	49	(3)
Handling the sum or difference of cubes	50	(1)
Tackling quadratic-like trinomials	51	(1)
Solving Quadratic Inequalities	52	(5)
Keeping it strictly quadratic	53	(1)
Signing up for fractions	54	(1)
Increasing the number of factors	55	(2)
Rooting Out the Rational, Radical, and Negative	57	(20)
Acting Rationally with Fraction-Filled Equations	57	(8)
Solving rational equations by tuning in your LCD	58	(4)
Solving rational equations with proportions	62	(3)
Ridding Yourself of a Radical	65	(3)
Squaring both sides of a radical equation	65	(2)
Calming two radicals	67	(1)
Changing Negative Attitudes about Exponents	68	(5)
Flipping negative exponents out of the picture	69	(1)
Factoring out negatives to solve equations	70	(3)
Fooling Around with Fractional Exponents	73	(4)
Combining terms with fractional exponents	73	(1)
Factoring fractional exponents	73	(1)
Solving equations by working with fractional exponents	74	(3)
Graphing Your Way to the Good Life	77	(20)
Coordinating Your Graphing Efforts	78	(2)
Identifying the parts of the coordinate plane	78	(1)
Plotting from dot to dot	79	(1)
Streamlining the Graphing Process with Intercepts and Symmetry	80	(4)
Finding x-and y-intercepts	80	(2)
Reflecting on a graph's symmetry	82	(2)
Graphing Lines	84	(5)
Finding the slope of a line	85	(1)
Facing two types of equations for lines	86	(2)
Identifying parallel and perpendicular lines	88	(1)
Looking at 10 Basic Forms	89	(4)
Lines and quadratics	90	(1)
Cubics and quartics	90	(1)
Radicals and rationals	91	(1)
Exponential and logarithmic curves	92	(1)
Absolute values and circles	93	(1)
Solving Problems with a Graphing Calculator	93	(4)
Entering equations into graphing calculators correctly	94	(2)
Looking through the graphing window	96	(1)
Part II: Facing Off With Functions	97	(104)
Formulating Function Facts	99	(18)
Defining Functions	99	(2)
Introducing function notation	100	(1)
Evaluating functions	100	(1)
Homing In on Domain and Range	101	(3)
Determining a function's domain	101	(1)
Describing a function's range	102	(2)
Betting on Even or Odd Functions	104	(2)
Recognizing even and odd functions	104	(1)
Applying even and odd functions to graphs	105	(1)
Facing One-to-One Confrontations	106	(2)
Defining one-to-one functions	106	(1)
Eliminating one-to-one violators	107	(1)
Going to Pieces with Piecewise Functions	108	(3)
Doing piecework	108	(2)
Applying piecewise functions	110	(1)
Composing Yourself and Functions	111	(3)
Performing compositions	112	(1)
Simplifying the difference quotient	113	(1)
Singing Along with Inverse Functions	114	(3)
Determining if functions are inverses	114	(1)
Solving for the inverse of a function	115	(2)
Sketching and Interpreting Quadratic Functions	117	(16)
Interpreting the Standard Form of Quadratics	117	(3)
Starting with ``a'' in the standard form	118	(1)
Following up with ``b'' and ``c''	119	(1)
Investigating Intercepts in Quadratics	120	(4)
Finding the one and only y-intercept	120	(2)
Finding the x-intercepts	122	(2)
Going to the Extreme: Finding the Vertex	124	(2)
Lining Up along the Axis of Symmetry	126	(1)
Sketching a Graph from the Available Information	127	(2)
Applying Quadratics to the Real World	129	(4)
Selling candles	129	(1)
Shooting basketballs	130	(1)
Launching a water balloon	131	(2)
Staying Ahead of the Curves: Polynomials	133	(24)
Taking a Look at the Standard Polynomial Form	133	(1)
Exploring Polynomial Intercepts and Turning Points	134	(5)
Interpreting relative value and absolute value	135	(1)
Counting intercepts and turning points	136	(1)
Solving for polynomial intercepts	137	(2)
Determining Positive and Negative Intervals	139	(3)
Using a sign-line	139	(2)
Interpreting the rule	141	(1)
Finding the Roots of a Polynomial	142	(7)
Factoring for polynomial roots	143	(2)
Saving your sanity: The Rational Root Theorem	145	(3)
Letting Descartes make a ruling on signs	148	(1)
Synthesizing Root Findings	149	(8)
Using synthetic division to test for roots	150	(3)
Synthetically dividing by a binomial	153	(1)
Wringing out the Remainder (Theorem)	154	(3)
Relying on Reason: Rational Functions	157	(20)
Exploring Rational Functions	158	(1)
Sizing up domain	158	(1)
Introducing intercepts	159	(1)
Adding Asymptotes to the Rational Pot	159	(5)
Determining the equations of vertical asymptotes	160	(1)
Determining the equations of horizontal asymptotes	160	(1)
Graphing vertical and horizontal asymptotes	161	(1)
Crunching the numbers and graphing oblique asymptotes	162	(2)
Accounting for Removable Discontinuities	164	(2)
Removal by factoring	164	(1)
Evaluating the removal restrictions	165	(1)
Showing removable discontinuities on a graph	165	(1)
Pushing the Limits of Rational Functions	166	(7)
Evaluating limits at discontinuities	168	(2)
Going to infinity	170	(2)
Catching rational limits at infinity	172	(1)
Putting It All Together: Sketching Rational Graphs from Clues	173	(4)
Exposing Exponential and Logarithmic Functions	177	(24)
Evaluating Exponential Expressions	177	(1)
Exponential Functions: It's All About the Base, Baby	178	(4)
Observing the trends in bases	179	(1)
Meeting the most frequently used bases: 10 and e	180	(2)
Solving Exponential Equations	182	(3)
Making bases match	182	(2)
Recognizing and using quadratic patterns	184	(1)
Showing an ``Interest'' in Exponential Functions	185	(4)
Applying the compound interest formula	185	(3)
Looking at continuous compounding	188	(1)
Logging On to Logarithmic Functions	189	(4)
Meeting the properties of logarithms	189	(1)
Putting your logs to work	190	(3)
Solving Logarithmic Equations	193	(3)
Setting log equal to log	193	(2)
Rewriting log equations as exponentials	195	(1)
Graphing Exponential and Logarithmic Functions	196	(5)
Expounding on the exponential	196	(2)
Not seeing the logs for the trees	198	(3)
Part III: Conquering Conies and Systems of Equations	201	(66)
Cutting Up Conic Sections	203	(22)
Cutting Up a Cone	203	(1)
Opening Every Which Way with Parabolas	204	(9)
Looking at parabolas with vertices at the origin	205	(3)
Observing the general form of parabola equations	208	(1)
Sketching the graphs of parabolas	209	(3)
Converting parabolic equations to the standard form	212	(1)
Going Round and Round in Conic Circles	213	(2)
Standardizing the circle	213	(1)
Specializing in circles	214	(1)
Preparing Your Eyes for Solar Ellipses	215	(4)
Raising the standards of an ellipse	216	(2)
Sketching an elliptical path	218	(1)
Feeling Hyper about Hyperbolas	219	(4)
Including the asymptotes	220	(2)
Graphing hyperbolas	222	(1)
Identifying Conies from Their Equations, Standard or Not	223	(2)
Solving Systems of Linear Equations	225	(22)
Looking at the Standard Linear-Systems Form and Its Possible Solutions	225	(1)
Graphing Solutions of Linear Systems	226	(3)
Pinpointing the intersection	227	(1)
Toeing the same line twice	228	(1)
Dealing with parallel lines	228	(1)
Eliminating Systems of Two Linear Equations with Addition	229	(3)
Getting to an elimination point	230	(1)
Recognizing solutions for parallel and coexisting lines	231	(1)
Solving Systems of Two Linear Equations with Substitution	232	(2)
Variable substituting made easy	232	(1)
Identifying parallel and coexisting lines	233	(1)
Using Cramer's Rule to Defeat Unwieldy Fractions	234	(3)
Setting up the linear system for Cramer	235	(1)
Applying Cramer's Rule to a linear system	236	(1)
Raising Linear Systems to Three Linear Equations	237	(4)
Solving three-equation systems with algebra	237	(2)
Settling for a generalized solution for linear combinations	239	(2)
Upping the Ante with Increased Equations	241	(2)
Applying Linear Systems to Our 3-D World	243	(1)
Using Systems to Decompose Fractions	244	(3)
Solving Systems of Nonlinear Equations and Inequalities	247	(20)
Crossing Parabolas with Lines	247	(4)
Determining the point(s) where a line and parabola cross paths	248	(2)
Dealing with a solution that's no solution	250	(1)
Intertwining Parabolas and Circles	251	(4)
Managing multiple intersections	252	(2)
Sorting out the solutions	254	(1)
Planning Your Attack on Other Systems of Equations	255	(9)
Mixing polynomials and lines	256	(1)
Crossing polynomials	257	(2)
Navigating exponential intersections	259	(2)
Rounding up rational functions	261	(3)
Playing Fair with Inequalities	264	(3)
Drawing and quartering inequalities	264	(1)
Graphing areas with curves and lines	265	(2)
Part IV: Shifting into High Gear with Advanced Concepts	267	(80)
Simplifying Complex Numbers in a Complex World	269	(12)
Using Your Imagination to Simplify Powers of i	270	(1)
Understanding the Complexity of Complex Numbers	271	(5)
Operating on complex numbers	272	(1)
Multiplying by the conjugate to perform division	273	(2)
Simplifying radicals	275	(1)
Solving Quadratic Equations with Complex Solutions	276	(2)
Working Polynomials with Complex Solutions	278	(3)
Identifying conjugate pairs	278	(1)
Interpreting complex zeros	279	(2)
Making Moves with Matrices	281	(22)
Describing the Different Types of Matrices	282	(2)
Row and column matrices	282	(1)
Square matrices	283	(1)
Zero matrices	283	(1)
Identity matrices	284	(1)
Performing Operations on Matrices	284	(8)
Adding and subtracting matrices	285	(1)
Multiplying matrices by scalars	286	(1)
Multiplying two matrices	286	(2)
Applying matrices and operations	288	(4)
Defining Row Operations	292	(1)
Finding Inverse Matrices	293	(6)
Determining additive inverses	294	(1)
Determining multiplicative inverses	294	(5)
Dividing Matrices by Using Inverses	299	(1)
Using Matrices to Find Solutions for Systems of Equations	300	(3)
Making a List: Sequences and Series	303	(20)
Understanding Sequence Terminology	303	(6)
Using sequence notation	304	(1)
No-fear factorials in sequences	304	(1)
Alternating sequential patterns	305	(1)
Looking for sequential patterns	306	(3)
Taking Note of Arithmetic and Geometric Sequences	309	(3)
Finding common ground: Arithmetic sequences	309	(2)
Taking the multiplicative approach: Geometric sequences	311	(1)
Recursively Defining Functions	312	(1)
Making a Series of Moves	313	(5)
Introducing summation notation	314	(1)
Summing arithmetically	315	(1)
Summing geometrically	316	(2)
Applying Sums of Sequences to the Real World	318	(4)
Cleaning up an amphitheater	318	(1)
Negotiating your allowance	319	(1)
Bouncing a ball	320	(2)
Highlighting Special Formulas	322	(1)
Everything You Wanted to Know about Sets	323	(24)
Revealing Set Notation	323	(4)
Listing elements with a roster	324	(1)
Building sets from scratch	324	(1)
Going for all (universal set) or nothing (empty set)	325	(1)
Subbing in with subsets	325	(2)
Operating on Sets	327	(3)
Celebrating the union of two sets	327	(1)
Looking both ways for set intersections	328	(1)
Feeling complementary about sets	329	(1)
Counting the elements in sets	329	(1)
Drawing Venn You Feel Like It	330	(6)
Applying the Venn diagram	331	(1)
Using Venn diagrams with set operations	332	(1)
Adding a set to a Venn diagram	333	(3)
Focusing on Factorials	336	(2)
Making factorial manageable	336	(1)
Simplifying factorials	337	(1)
How Do I Love Thee? Let Me Count Up the Ways	338	(6)
Applying the multiplication principle to sets	338	(1)
Arranging permutations of sets	339	(4)
Mixing up sets with combinations	343	(1)
Branching Out with Tree Diagrams	344	(3)
Picturing a tree diagram for a permutation	345	(1)
Drawing a tree diagram for a combination	346	(1)
Part V: The Part of Tens	347	(14)
Ten Multiplication Tricks	349	(8)
Ten Special Types of Numbers	357	(4)
Index	361