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THE ESSENTIALS ®OF PRE-CALCULUS
Copyright © 1992 by Research and Education Asso-ciation. All rights reserved. No part of this book maybe reproduced in any form without permission of thepublisher.
Printed in the United States of America
Library of Congress Catalog Card Number 91-62041
International Standard Book Number 0-87891-877-9
REVISED PRINTING, 1993
ESSENTIALS is a registered trademark ofResearch and Education Association, Piscataway, New Jersey 08854
WHAT "THE ESSENTIALS"WILL DO FOR YOU
This book is a review and study guide. It is comprehen-sive and it is concise.
It helps in preparing for exams, in doing homework, andremains a handy reference source at all times.
It condenses the vast amount of detail characteristic of thesubject matter and summarizes the essentials of the field.
It will thus save hours of study and preparation time.
The book provides quick access to the important facts,principles, theorems, concepts, and equations in the field.
Materials needed for exams can be reviewed in summaryform - eliminating the need to read and re-read many pagesof textbook and class notes. The summaries will even tend tobring detail to mind that had been previously read or noted.
This "ESSENTIALS" book has been prepared by anexpert in the field, and has been carefully reviewed to assureaccuracy and maximum usefulness.
Dr. Max FogielProgram Director
in
CONTENTS
Chapter No. Page No.
1 SETS, NUMBERS, OPERATIONS, ANDPROPERTIES 1
1.1 Sets 11.2 Real Numbers and Their Components 21.3 Real Number Properties of Equality 31.4 Real Number Operations and Their Properties 31.5 Complex Numbers 6
2 COORDINATE GEOMETRY 82.1 Real Numbers and the Coordinate Line 82.2 Cartesian Coordinates 92.3 Lines and Segments 102.4 Symmetry 132.5 Intercepts and Asymptotes 162.6 Parametric Equations 172.7 Polar Coordinates 18
3 FUNDAMENTAL ALGEBRAIC TOPICS 203.1 The Arithmetic of Polynomials 203.2 Factoring 213.3 Rational Expressions 223.4 Radicals 24
4 SOLVING EQUATIONS AND INEQUALITIES 274.1 Solving Equations in One Variable 274.2 Solving Inequalities 284.3 Solving Absolute Value Equations and Inequalities 294.4 Solving Systems of Linear Equations 31
IV
4.5 Solving Systems of Linear Inequalities andLinear Programming 35
4.6 Solving Equations Involving Radicals, Solving RationalEquations, and Solving Equations in Quadratic Form 37
5 FUNCTIONS 405.1 Functions and Equations 405.2 Combining Functions 425.3 Inverse Functions 445.4 Increasing and Decreasing Functions 465.5 Polynomial Functions and Their Graphs 465.6 Rational Functions and Their Graphs 495.7 Special Functions and Their Graphs 50
6 TRIGONOMETRY 536.1 Trigonometric Functions 536.2 Inverse Trigonometric Functions 596.3 Trigonometric Identities 596.4 Triangle Trigonometry 606.5 Solving Trigonometric Equations 626.6 The Trigonometric Form of a Complex Number 63
7 EXPONENTS AND LOGARITHMS 667.1 Integer Exponents 667.2 Rational Number Exponents 677.3 Exponential Functions 677.4 Logarithms 687.5 Logarithmic Functions 71
8 CONIC SECTIONS 738.1 Introduction to Conic Sections 738.2 The Circle 748.3 The Ellipse 758.4 The Parabola 778.5 The Hyperbola 79
9 MATRICES AND DETERMINANTS 829.1 Matrices and Matrix Operations 829.2 Determinants 839.3 The Inverse of a Square Matrix 86
10 MISCELLANEOUS TOPICS 8910.1 The Fundamental Counting Principle, Permutations, and
Combinations 8910.2 The Binomial Theorem 9010.3 Sequences and Series 9010.4 Mathematical Induction 9210.5 Zeros of a Polynomial 93
VI
CHAPTER 1
SETS, NUMBERS, OPERATIONS,AND PROPERTIES
1.1 SETSA collection of objects categorized together is called a set.
There are two standard ways to represent sets, the roster method andthe set-builder method. For example,
{2, 3, 4}is in roster form and
{x | x is a counting number between 1 and 5}is in set-builder form, although both describe the same set.
The symbol "G" is used to represent "is an element of and thesymbol "£" is used to represent "is a subset of." Here are two impor-tant definitions concerning sets.
A C B if and only if every element of A is an element of B.A = B if and only if A C B and B £ A.
Two important set operations are union, denoted by "U," andintersection, denoted by "PI," defined below.
1.2 REAL NUMBERS AND THEIR COMPONENTS
Real numbers provide the basis for most precalculus math-ematics topics. The set of all real numbers has various components.These components are the set of all natural numbers, N, the set of allwhole numbers, W, the set of all integers, /, the set of all rationalnumbers, Q, and the set of all irrational numbers, S. Then,
#={1,2,3,...},
W= {0,1,2,3,...},
/ ={...,-3,-2,-1,0, 1,2,3, ...},
Q = {"/b\a,bEI and b * 0},
and 5 = { x \ x has a decimal name which is nonterminating anddoes not have a repeating block}.
It is obvious that N C W, W C I, and / C Q, but a similar rela-tionship does not hold between Q and 5. More specifically, the deci-mal names for elements of Q are
(1) terminating or
(2) nonterminating with a repeating block.
For example, ]/2 = -5 and Va = .333.... This means that Q and 5 have nocommon elements. Examples of i rrat ional numbers include.101001000...,31, and V2.
All real numbers are normally represented by R and R =Q U S. This means that every real number is either rational or irratio-nal. A nice way to visualize real numbers geometrically is that realnumbers can be put in a one-to-one correspondence with the set ofall points on a line.
1.3 REAL NUMBER PROPERTIES OF EQUALITYThe standard properties of equality involving real numbers are:
Reflexive Property of EqualityFor each real number a,
a- a.
Symmetric Property of EqualityFor each real number a, for each real number b,
if a = b, then b = a.
Transitive Property of EqualityFor each real number a, for each real number b, for each realnumber c,
if a = b and b = c, then a = c.
Other properties of equality are listed in Chapter 4.
1.4 REAL NUMBER OPERATIONS AND THEIRPROPERTIESThe operations of addition and multiplication are of particular
importance. As a result many properties concerning those operationshave been determined and named. Here is a list of the most importantof these properties.
Closure Property of AdditionFor every real number a, for every real number b,
a + b
is a real number.
Closure Property of MultiplicationFor every real number a, for every real number b,
ab
is a real number.
Commutative Property of AdditionFor every real number a, for every real number b,
a + b = b + a.
Commutative Property of MultiplicationFor every real number a, for every real number b,
ab = ba.
Associative Property of AdditionFor every real number a, for every real number b, for every realnumber c,
(a + b) + c = a + (b + c).
Associative Property of MultiplicationFor every real number a, for every real number b, for every realnumber c,
(ab)c = a(bc).
Identity Property of AdditionFor every real number a,
a + 0 = 0 + a = a.
Identity Property of MultiplicationFor every real number a,
a • 1 = 1 • a = a.
Inverse Property of AdditionFor every real number a, there is a real number - a such that
a + -a = -a + a = 0.
Inverse Property of MultiplicationFor every real number a, a * 0, there is a real number or1 suchthat
a • a~} = a-1 • a = 1.
Distributive PropertyFor every real number a, for every real number b, for every realnumber c,
a(b + c) - ab + ac.
The operations of subtraction and division are also important,but less important than addition and multiplication. Here are thedefinitions for these operations.
For every real number a, for every real number b, for every realnumber c,
a - b = c if and only if b + c - a.
For every real number a, for every real number b, for every realnumber c,
a + b = cif and only if c is the unique real number such thatbe = a.
The definition of division eliminates division by 0. Thus, for example,4 + 0 is undefined, 0 + 0 is undefined, but 0 + 4 = 0.
In many instances, it is possible to perform subtraction by firstconverting a subtraction statement to an addition statement. This isillustrated below.
For every real number a, for every real number b,
In a similar way, every division statement can be converted to amultiplication statement. Use the following model:
For every real number a, for every real number b,b#Q,
a + b = a • b~l.
1.5 COMPLEX NUMBERSAs indicated above, real numbers provide the basis for most
precalculus mathematics topics. However, on occasion there arisesituations in which real numbers by themselves are not enough toexplain what is happening. As a result, complex numbers developed.
Returning momentarily to real numbers, the square of a realnumber cannot be negative. More specifically, the square of a positivereal number is positive, the square of a negative real number ispositive, and the square of 0 is 0. Then i is defined to be a numberwith a property that
i2 -
Obviously, i is not a real number. C is then used to represent the setof all complex numbers and
C = { a + bi \ a and b are real numbers }.
Here are the definitions of addition, subtraction, and multiplicationof complex numbers.
Suppose x + yi and z + wi are complex numbers. Then
(x + yi) + (z + wi) = (x + z) + (y + w)i
(x + yi) - (z + wi) = (x - z) + (y - w)i
(x + yi) x (z + wi) = (xz - y w) + (xw + yz)i.
Division of two complex numbers is usually accomplished with aspecial procedure that involves the conjugate of a complex number.The conjugate of a + bi is denoted by
a + bi and a + bi = a - bi.
Also, (a + bi) (a - bi) = a2 + b2.
The usual procedure for division is illustrated below.
x + yiz + wi
x + yt z-wiZ + Wl Z-Wl
(xz + yw) + (-xw + yz)i
All the properties of real numbers described in the previous sectioncarry over to complex numbers, however, those properties will notbe stated again.
If a is a real number, then a can be expressed in the form a =a + oi. Hence, every real number is a complex number andRCC.
CHAPTER 2
COORDINATE GEOMETRY
2.1 REAL NUMBERS AND THE COORDINATE LINEAs was mentioned in Section 1.2, real numbers can be put in a
one-to-one correspondence with the points on a line. The commonpractice is to select a point, call it the origin, and assign to it thenumber 0. Then the positive real numbers are identified with thepoints to the right of the origin, and the negative real numbers areassociated with the points to the left of the origin.
- 4 - 3 - 2 - 1 0 1 2
Figure 2.13 4
If Pj and P2 are associated with the real numbers *i and x2, then thedistance (d) between PI and P2 is defined to be
d = \x2-xl\An uninterrupted portion of a number line is called an interval.
The symbol (a, b) is called an open interval and is defined below.
(a, b) = { x | x is a real number and a < x < b} I
The graph of this interval is shown below.
a b
Figure 2.2In this graph, the open circles at the two ends indicate that the pointsassociated with a and b are not included in the interval.
The symbol [a, b] is called a closed interval, and its definitionand graph follow.
[a, b] = { x | x is a real number and a s x s b}
Figure 2.3In this instance, the closed dots indicate that the points associatedwith a and b are included in the interval. Similarly, (a, b] and [a, b) arecalled half open intervals and
(a, b] = { x | x is a real number and a < x s b}
[a, b) = {x | x is a real number and a s x < b}.
2.2 CARTESIAN COORDINATESIn a coordinate system, a specific horizontal line is called the
A:-axis, and a specific vertical line is called the y-axis. The point of in-tersection of these lines is called the origin. (See Figure 2.4.)
By starting at the origin, one way to get to point P is to movetwo units to the left and then move one unit up. Then P is associatedwith (- 2, 1), and (- 2, 1) are called the coordinates of P. More spe-cifically, - 2 is the x coordinate of P and 1 is the y-coordinate of P. Theaxes divide the plane into four quadrants which are labeled I, II, III,and IV, as indicated in the illustration below. It should be noted that(a, b) has one meaning in Cartesian coordinates and another dis-tinctly different meaning in the coordinate geometry of a line.
Quadrant II '
P*
~* ——— I ——— 1_ _. ,
' y Quadrant 1
- 2
- 1
„ _| — ..,.„„„-2 -1
- - -1
Quadrant III
- ~ — 2
Figure 2.4
Quadrant IV
The notation P(x, y) means that P is a point whose coordinatesare (x, y).
2.3 LINES AND SEGMENTSUsing the Pythagorean theorem, it is fairly easy to establish
that the distance between Pfa, yi) and Q(x2, y2) is
Also, the coordinates of the midpoint of the segment joining P(xl,and Q(x2, y2) are
+ x
The slope of a line is a number which indicates the slant, ordirection, of the line. The symbol "m" is used to represent the slopeof a line. For a nonvertical line which passes through Pfa, y^ andC(*2> y& the slope is
m =x2-xl
10
From properties of similar triangles, it is easy to show that theslope of a nonvertical line is unique and its calculation is indepen-dent of the two points selected. It is somewhat common to use thesymbol Ay to represent y2 -y\ and the symbol Ax to represent x2 - x\.Using these symbols,
m -Ax
A good intuitive interpretation of slope is very important incalculus and precalculus mathematics. Lines with various slopes arepictured below.
This line has no slope.j \
Figure 2.5It can be determined whether or not two lines are parallel or
perpendicular by inspecting their slopes. Here is how this can bedone when lines /t and /2 have slopes m\ and m2.
/! || /2 »f and only if mi = m2
/, 1 /2 if and only if ml • m2 = - 1Since any two vertical lines are parallel, and since a slope does notexist for a vertical line, when two lines do not have a slope they areparallel. Also, since any vertical line is perpendicular to any horizon-
11
tal line, any line with slope 0 is perpendicular to any line which doesnot have a slope.
Suppose points P(0, b) and Q(a, 0) are points on a line. Then, ais called the ̂ -intercept of the line and b is called the ̂ -intercept of theline. An equation of the form
a b
is said to be in intercept form. Here is the graph of this line.
y 4
' (a, 0)/
/
(0,6)
Figure 2.6An equation of the form
y = mx + b
is said to be in the slope-intercept form. Here is a graph of this line., y
(0,6)
The slope of this line is m.
Figure 2.7
12
If (xi, y\) is a point on a line, then
is said to be in point-slope form. Here is a graph of this line.
The slope of this line is m.
Figure 2.8It is somewhat common for someone to mention the equation of a line.This is, of course, incorrect. A line has many equations but the onesmentioned above are important enough to be given specific names.
The distance between a point and a line refers to the shortestpossible distance between the point and the line. This is, of course,the perpendicular distance between the point and the line. For a pointP(x0, y0) and a line with slope m and y-intercept b, this distance isgiven by
\l
2.4 SYMMETRYTwo points PI and P2 are said to be symmetric with respect to
line /, if / is the perpendicular bisector of PjP2 .(See Figure 2.9.)
A figure is symmetric with respect to line / if for every point Pton the figure there is another point P2 on the figure such that PI and P2are symmetric relative to /. Intuitively, this means that one portion ofthe figure is the mirror image of the other portion of the figure. (SeeFigure 2.10.)
13
Figure 2.9For the figure below, / is a symmetry line.
Figure 2.10Two points F] and F2
are symmetric with respect to Q when Qis the midpoint of P\Pf A figure is said to be symmetric relative to apoint Q if for every point P\ on the figure, there is a point P2
on tne
figure such that PI and P2 are symmetric relative to Q.
Figure 2.11
For the figure above, Q is a symmetry point.
When graphing equations, it is often valuable to know aboutsymmetry points and lines. It is particularly important to know whethera given equation will have a graph which is symmetric to the origin,the *-axis, the y-axis, and the line y = x. Here is a table which de-
14
scribes techniques for finding lines and points of symmetry.
Replace
II the resulting equation Is equivalent to theoriginal equation, then the figure has symmetrywith respect to the
(x,y)by(-x,-y) origin
(x,y)by(-x,y) y-axis
(x, y) by (y, x) line)' = x
Table 2.1Consider the equation y = x2. Since (- x)2 = x2, the y-axis is a line
of symmetry for the graph of this equation. The graph of this equationfollows.
Figure 2.12Next, consider the equation y - x + x3. Then, - y = - x + (- x)3 isequivalent to y = x + jc3, and the origin is a point of symmetry for thecorresponding graph. The graph follows (Figure 2.13).
15
Figure 2.13
2.5 INTERCEPTS AND ASYMPTOTESThe ^-intercept and v-intercept for a line were described in
Section 2.3. These terms have a similar meaning for the graph of acurve. Specifically, the second coordinates of points where a curvecrosses the y-axis are called the y-intercepts, and the first coordinatesof points where the curve crosses the jc-axis are called the *-inter-cepts.
" y
,<*2 .o)
Figure 2.14For the curve pictured above, the ^-intercepts are x\, x2, and x3 whilethe only y-intercept is yj.
When the graph of an equation approaches a line, but nevertouches it, the line is said to be an asymptote. In most cases, theselines are vertical or horizontal and are called vertical asymptotes and
16
horizontal asymptotes. In the picture below, the dotted lines are as-ymptotes. »
Figure 2.15
2.6 PARAMETRIC EQUATIONSThe usual method for describing a curve is to give a single
equation relating the variables x and y. However, sometimes it is moreconvenient to use two equations that express x and y in terms of a thirdvariable. In such a case, the third variable is called a parameter andthe equations are called parametric equations. For example,
x = cos 0 y = sin 9
is an example of parametric equations with 0 as the parameter. It isquite easy to show that the corresponding graph is a circle and, morespecifically, the equations are equivalent to the equation
x2 + y2 = I2.
A cycloid is defined to be the curve traced by a fixed point ona circle when the circle rolls on a line. With appropriate choice of thecoordinate system, here is the graph resulting from rolling a circle ofradius a. (See Figure 2.16.)
The parametric equations for the cycloid are
x = a(0 - sin 6) and y = a(l - cos 0),
17
Figure 2.16where 6 must be measured in radians. It is possible to eliminate theparameter 9 by a substitution method, but the resulting equation isquite complicated and so for this reason, the parametric equations areused almost invariably for a cycloid.
2.7 POLAR COORDINATESIn the rectangular coordinate system, a point is located by its
distances from the *-axis and the y-axis. In the polar coordinate sys-tem, a fixed point O is selected as the origin and a fixed half line OAis chosen as the polar axis. Then the polar coordinates of a point Pare (r, 9), where 9 is the measure of the angle that OP makes with OA,and r is the distance from O to P.
O A
Figure 2.17Then, when graphing the point P(r, 9)
18
f (1) using the polar axis as the initial side, lay off an angle ofmeasure 6 (counterclockwise if 8 is positive and clockwise if 6is negative), and
(2) on the terminal side of this angle, measure off a segment oflength r, if r is positive, and if r is negative, extend the termi-nal side of the angle past O and measure a segment of length| r | along the extended side.
In practice, graphing in the polar coordinate system is usually doneon special polar coordinate graph paper. An example follows.
JT2
r = 2 + 2 cos 9
Figure 2.18The heart-shaped figure is the graph of r = 2 + 2 cos 9 and is called acardioid.
19
CHAPTER 3
FUNDAMENTAL ALGEBRAIC TOPICS
3.1 THE ARITHMETIC OF POLYNOMIALSA monomial is a real number, or the product of a real number,
with one or more variables having whole number exponents. A poly-nomial is a finite sum of monomials. The sum or difference of twopolynomials is found by combining like terms. The product of twopolynomials can be determined by using the distributive property.However, here are some products which should be remembered.
2.
3.
4.
5.
6.
1.
8.
(x+y)(x-y) = x2-y2
(x + y)2 = x2 + 2xy + y2
(x-y)2 = x2-2xy+y2
(x + y)3 = x3 + 3x2y + 3xy2 + y3
(x + y) (x2 - xy + y2) = x3 + y3
(ax + b) (ex + d) = acx2 + (ad + bc)x + bd
The process for finding the quotient of two polynomials is somewhatmore complicated. This process is illustrated with the example below.
20
EXAMPLEFind
2x2 -lx + 8x-2
2x -3
x-2 2x2 -lx + 82x2-4x
-3x +-3jc +
8
Thus,
2x2 - Ix•x-2 x-2
3.2 FACTORINGFactoring is the inverse operation of finding products. In this
situation, a polynomial is given and the student is to express thepolynomial as the product of at least two polynomials. In the process,it is imperative that the student memorize the product types given inthe previous section. Here are some procedures which make the pro-cess easier.
(1) Look for and factor out any common monomial factors.
(2) Look for any of the product types and apply any identifiedpatterns.
(3) Try a trial-and-error approach.
(4) Try grouping terms differently.
Here are some examples which illustrate these procedures.
21
EXAMPLEFactor
X2-16
Factor
4x2 +
4x2 + \2xy + 9y2 = (2x + 3y)2
Factor
2x3 - Icy3 = 2(x3 - 8y3)
Factor
2x2 - xy - 6y2
2x2-xy-6y2 =
Factor
ax + ay + 3x + 3ax + ay + 3x +
3.3 RATIONAL EXPRESSIONS
A rational expression is an expression of the type ^y{, where
3y) (x - 2y)
= a(x + y) + 3(x + y)
P(x)G(*)'
P(x) and Q(x) are both polynomials in the variable x and Q(x) »* 0.
Also, P(x)—*—L is in simplest form whenever the only common factorsGW
of P(x) and Q(x) are 1 and - 1.
22
Here are some important generalizations concerning rationalexpressions.
**(*) *(*)Q(x) S(*)
if and only if P(x) • S(x) = Q(x) • R(x)
Q(x) Q(x)-S(x)
GOO Q(x) Q(x)P(x) R(x) _P(x)-R(x)Q(x) Q(x)~ Q(x)P(x± RJx± P(x)-R(x)Q(x) S(x) = Q(x)-S(x)P& + R&=P(x± S(x)Q(x) S(x) Q(x) R(x)
Here are some examples which illustrate the application of thesegeneralizations.
EXAMPLESimplify
x2+3x-W x2-2x-3x2-4
x2-4 x+2x-l55)(*-2)Qc-3)(;c + l
x + lx + 2
23
Simplify
x-2 x + 3 -3x2 -1 *2+3* + 2 x2+x-2x-2 _ x + 3 _ -3x2 -1 ~ x2 + 3x + 2 ~ x2 + x - 2
x-2 x+3
(x-2}(x + 2) _ (x + 3) (x -2)(x - l)(x + l)(x + 2) (x + 2)(x + l)(x - 1) 2)(x-l)(x(x
2 _ 4) _ + 2* - 3) - (-3x - 3))(* + !)(* + 2)
x + 2
3.4 RADICALS
The symbol "la" is used to represent the principle nlh root of aand more specifically " \ 4 " is used to represent the principal squareroot of 4. Since, for example,
22 = 4 and (- 2)2 = 4,
2 and - 2 are candidates for v 4 . Thus, there is need for a definitionwhich differentiates between these two options. A similar situationoccurs for any even root. Also, since a real number raised to an evenpower cannot be negative, 1 a is not a real number when n is even anda is negative. When n is odd la is meaningful for any real numbera. Thus, there is need to differentiate between the even and odd casesfor la . Here are the definitions for t a .
For every real number a, a a 0, and for an even positive integern,
24
1 a = b if and only if b" = a and b a 0.
For every real number a, for every odd positive integer n,
n> I , la = b if and only if b" = a.
Here are several important generalizations concerning radicals.
If a and b are real numbers, and if n is a positive integer suchthat 1'fl and 16 are real numbers, then
la" = a (when « is odd)
la" =|a| (when n is even)
These generalizations can be used to rewrite radical expressions indifferent ways. In this regard, a radical expression is said to be instandard (simplest) form if
(1) there are no polynomial factors in the radicand raised to ahigher power than the index of the radical.
(2) there are no fractions under the radical sign.
(3) there are no radicals contained in the denominator.
(4) the index of the radical is as small as possible.
Here are some examples where radicals are transformed to standardform.
EXAMPLE
25
Nl - (4- \3)
(4 + \3)(4-V3)
4-V316-3
4-V313
Notice that in the last example, the generalization l a - a1'" isused. This generalization is discussed in Section 7.2.
26
CHAPTER 4
SOLVING EQUATIONSAND INEQUALITIES
4.1 SOLVING EQUATIONS IN ONE VARIABLEThere are a number of procedures which can be used in equation
solving situations. Here are generalizations which are helpful in thisregard.
(1) For every real number a, for every real number b, for every realnumber c,
a = b if and only if a + c = b + c
a = b if and only if ac = be
a = b if and only if a- c = b-c
a = b if and only if a/c = b/c
(2) If a and b are real numbers and ab = 0, then
a = 0 or b = 0.
(c-0)
(c-0)
Here is an example which illustrates these procedures.
EXAMPLESolve
27
2x2-x-l = 0
(2r + l)(x-l) = 0
2x + 1 = 0 or x-l=0
x = -l/2 or A: = 1
The famous quadratic formula follows.
If a, b, and c are real numbers, a * 0, and a*2 + fcc + c = 0, then
-b±\b2 -4ac2a
The following example illustrates the application of this for-mula.
EXAMPLESolve
x =-3±\9-4-l(-2)
-3±\'17
4.2 SOLVING INEQUALITIESHere are some generalizations about inequalities which are
helpful in the solution of inequalities.
(1) For every real number a, for every real number b, for every realnumber c,
a < b if and only ifa + c < 6 + c
a < b if and only if a- c <b- c
(2) For every real number a, for every real number b, for every realnumber c, c> 0,
28
a < b if and only if ac < be
a < b if and only if a/c < blc
(3) For every real number a, for every real number b, for every realnumber c, c < 0,
a < b if and only if ac > be
a < b if and only if a/c > b/c
Comparable generalizations could be made about ">." More specifi-cally, if in the generalizations above, every "less than" symbol isreplaced by a "greater than" symbol and every "greater than" symbolis replaced by a "less than" symbol, appropriate generalizations willresult. Here is an example which illustrates the use of these generali-zations.
EXAMPLESolve
(- 2x + 3) + (- 3) < 8 + (- 3)
-2x<5
Notice that in the next to the last step, when both sides of the inequalityare multiplied by a negative number, the inequality changes from"less than" to "greater than."
4.3 SOLVING ABSOLUTE VALUE EQUATIONS ANDINEQUALITIESAbsolute value concepts occur often in problems in precalculus
mathematics and also in calculus. Here is the definition of absolutevalue.
29
For every real number x,
f x if x*0\-x if x<0
Here are the three generalizations which are valuable in thesolution of equations and inequalities involving absolute value con-cepts.
For every real number a, a > 0,
| x | = a if and only if * = a or x = - a
| x | < a if and only if - a < x < a
| x | > a if and only if * > a or x < - a
Here are three examples which illustrate appropriate use ofthese generalizations.
EXAMPLESolve
x = 5 or
Solve
= -5
- 3 < j t - 2 < 3
- l < ; c < 5
Solve
|x + 3 | > 4
x + 3 > 4 or * + 3 < - 4
x> 1 or x<-l
In the generalizations above, the requirement that a > 0 is significant.When that is not the case, intuitive methods can be used. For ex-
30
I x I < - 4 has no solution, while | x \ > - 4 has every real number as asolution.
4.4 SOLVING SYSTEMS OF LINEAR EQUATIONSConsider the system of linear equations listed below.
a^x + y = c2
Because the graphs of each of these equations are both straight lines,there are three geometric possibilities. One possibility is that thereare two distinct and intersecting lines. This is illustrated below.
y
Figure 4.1
In this case, there is exactly one ordered pair which satisfies bothequations. The equations are linearly independent and consistent.
A second possibility is that the two lines are parallel. This isillustrated in Figure 4.2 below.
Obviously, the two lines have no points in common, so theequations have no common solution. In this case, the equations aresaid to be linearly independent and inconsistent.
Finally, the two equations are equations of the same line. Thisis illustrated in Figure 4.3 below.
31
Figure 4.2y
Figure 4.3
Obviously, the two equations have an infinite number of commonsolutions. More specifically, any solution of either equation is a solu-tion of the other equation. In this case, the equations are said to belinearly dependent and consistent.
There are several well-known techniques for solving systemsof linear equations, three of which will be illustrated here. One of themost interesting techniques for solving systems of linear equations iscalled Cramer's Rule. This rule involves the evaluation of determinants(determinants are reviewed in Section 9.2).
Cramer's Rule (In the Case of Two Equations)Concerning the following two linear equations,
= c2
32
If,and
C2
and D x 0, thenDj_
X = D and y ••D
When D = 0, Dx x 0, and D,, * 0, the system of equations is linearlyindependent and inconsistent while, when D = Dx = Dy = 0, the sys-tem of equations is said to be linearly dependent and consistent. Hereis an example which illustrates the use of Cramer's Rule to solve asystem of two equations in two unknowns.
EXAMPLESolve the following system of equations using Cramer's Rule.
x + y = 1
2x-y = 5
D =1 12 - 1 '
= -3
1 1* ~ 5 -1
= -6Dv Dvx = -— — v = — *-D D
= 2 =-1
and Dv1 12 5
= 3
Notice that the determinant Dx is obtained by replacing the x-coeffi-cients in D by the corresponding constant terms, and Dy is obtained byreplacing the y-coefficients in D by the constant terms. Cramer's Rulealso applies in the case of n equations in n unknowns. More specifi-cally, with three equations and three unknowns,
Dv A, D,y - —^-, and z = •D D D
33
Inverses of MatricesSolutions of systems of equations can be determined using the
inverses of matrices. For example, the system of equations
a-iX + b^y = c2
can be represented by
AX = B
r«i &i32 2̂
where A B =
Then, and
Of course .A~1 is the multiplicative inverse of A. Once X is found, it iseasy to find x and y. This technique works for any system of n linearequations in n unknowns as long as A-' exists. The finding of inversesof square matrices is reviewed in Section 9.3. Here is an examplewhich illustrates the use of matrices to solve a system of two equationsin two unknowns.
EXAMPLESolve the following system of equations using the inverse of a
matrix.
AC + y = 1
2x-y = 5
n iA = 2 -1 t -i2. _1 53 3
x = 2 and y = - I
34
The Method of SubstitutionThe method of substitution uses one equation to solve for one
variable in terms of the other. This variable is then substituted in thesecond equation to have one equation using one variable. Solve forthis variable and then use it to obtain the other variable by virtue ofthe first equation.
EXAMPLE3x-y = 2
x-2y = l
Solve for_y from equation (1) in terms of x\
y = 3x-2;
use this in equation (2):
x-2(3x-2)= 1;
solve for x:
- 5x = - 3 => x = 3/5;
(1)
(2)
4.5 SOLVING SYSTEMS OF LINEAR INEQUALITIESAND LINEAR PROGRAMMINGIn the previous section, several approaches were given to the
solution of systems of linear equations. In this section, systems oflinear inequalities will be examined from a geometric viewpoint. Forexample, for specific real numbers a, b, and c, the graph of
ax + by + c = 0
is a line, while the graph of
ax + by + c < 0
is the half plane on one side of the line, with the graph of
35
ax + by + c > 0
being the half plane on the other side of the line. Consider the systemof inequalities below.
x + y z 6
x-y a -2
The solutions for this system are the set of all points in the shadedregion (including the boundary points).
(0,2)
(2, 4)
(6, 0)
Figure 4.4
In a linear programming problem, an objective function
C = ax + by + c
is given subject to constraints specified by linear inequalities. Then,if C assumes a maximum (or minimum), this maximum (minimum)must occur at a vertex for the system of linear inequalities. Suppose
and the objective is to maximize or minimize C subject to the systemof inequalities given above. A table is helpful
36
Vertex Corresponding value of C
(0,0) 2 - 0 + 0 = 0
(0,2) 2 - 0 + 2 = 2
(2,4) 2 - 2 + 4 = 8
(6,0) 6 - 2 + y = 12
Table 4.1Thus, the maximum value for C occurs at (6,0) while the minimumvalue occurs at (0,0).
4.6 SOLVING EQUATIONS INVOLVING RADICALS,SOLVING RATIONAL EQUATIONS, AND SOLVINGEQUATIONS IN QUADRATIC FORMIn solving equations involving radicals, it is sometimes necessary
to raise both sides of an equation to the same positive integral power.In this regard, the following generalization is applied:
If P(x) and Q(x) are expressions in x, and n is a positive integer,then the set of all solutions of the equation P(x) = Q(x) is asubset of the set of all solutions of
[/>(*)]" = [Q(x)]».
This means that when both sides of an equation are raised to thesame power, the resulting equation may have solutions which are notsolutions of the original equation. Thus, it is necessary to check tosee if solutions of the new equation are actually solutions of theoriginal equation. This is illustrated in the example below.
EXAMPLESolve
Vx + 2 - x = - 4
37
\x x-4
- 4)2
7 or x = 2
\7 + 2 -7 = -4 but
Thus, x = 1 is the only solution.
Equations often contain fractions that have denominators whichare polynomials. Such equations are called rational equations. Insolving equations of this type, it is often desirable to multiply bothsides of the equation by a polynomial to eliminate all fractions. Thismust be done cautiously. Here is a generalization which was statedoriginally in Section 4.1
For every real number a, for every real number b, for every realnumber c, c * 0.
a = b if and only if ac = be
The important idea is that c x 0. In practice, it is common to merelymultiply both sides by the appropriate polynomial and then check atthe end. Here is an example which illustrates this procedure.
EXAMPLESolve
xx-2
x
+ 1 = x2+4x2-4
x2 +4x2-4x-2
x(x + 2) + (x - 2) (x + 2) = x2 + 4
38
2x2 + 2x - 4 = x2 + 4
x2 +2*-8 = 0(x + 4)( je-2)-0x + 4 = 0 or *-2 =x = - 4 or x = 2
when * = - 4
- 4 - 2 (-4Y - 4
but when x = 2
2 2 2+ 4
2-2 2 2 -4
Thus, the only solution is x = - 4.
Equations of the form
ax2" + bx" + C = 0
are said to be in quadratic form. Equations of this type can often besolved by an appropriate substitution, as illustrated below.
EXAMPLESolve
Let y = x~l
6y2-y-2 =
3y - 2 = 0 or 2y + 1 = 0
y = 2/3 or y = - V2
jr» = 2/3 or r- 's-Vz
x = 3/2 or x = - 2
39
CHAPTER 5
FUNCTIONS
5.1 FUNCTIONS AND EQUATIONSThere are numerous examples in mathematics where a second
quantity is found from a first quantity by applying some rule. Considerthe equation
y = 2x + 3.
In this case, each y-value is determined from the x-value by multiply-ing the x-value by 2 and then adding 3 to the result. This is an exampleof a function. In general, a function is a rule which specifies anoutput number for each input number. Usually, the input numbers arespecified as .x-values and the output numbers are specified as y-val-ues. Here is the definition of the term function.
Let A and B be nonempty sets. Then a function from A to B is arule which assigns to each element of A exactly one element ofB. The domain of the function is A and the range consists onlyof those elements of B which are actually paired with elementsof ,4.
Pictures are often used to illustrate functions, and the symbol /is often used to represent a function. For the function/pictured below(Figure 5.1), the domain is A and
A = {a, b, c},
40
e
Figure 5.1while B = {d, e, J}, and the range is {d, e}. Notice that each elementof A is paired with exactly one element of B.
Many functions are merely illustrated in equation form. Forexample, y = 2x + 3 falls into that category. Unless it is specificallystated otherwise, the domain of this function is the set of all realnumbers. This same function can be denoted by
Also, ./(I) = 2 • 1 + 3
= 5.
This means that the number associated with 1 is 5 or, equivalently,y = 5 when* = 1.
When the function is denoted by an equation or by using thefix) notation, the domain is agreed to be the largest subset of realnumbers for which y or f(jt) is a real number. For example, the domainof
/(*)• x-\
is the set of all real numbers except 1, while the domain of y = \x isthe set of all nonnegative real numbers.
It is quite easy to tell whether or not a given graph is the graph
41
of a function. This is determined by the famous vertical line test.Consider the two graphs below.
A >
(a)
Figure 5.2Notice that for the graph on the left, each vertical line intersects thegraph in exactly one point. Thus, this is the graph of a function. Onthe other hand, for the graph on the right, there is a vertical linewhich intersects the curve in more than one point. In this case for agiven first component x\, there are two second components yt and y2and this is not the graph of a function. Here is a summary of theresults described above.
(1) When each vertical line intersects the graph in, at most, onepoint, the graph describes a function.
(2) When there exists a vertical line which intersects the graph inmore than one point, the graph does not describe a function.
5.2 COMBINING FUNCTIONSThere are various ways in which two real numbers can be
combined to form a third real number. For example, the sum, differ-ence, product, and quotient of two real numbers can be found. Simi-larly, two functions can be combined to form a third function and themost natural way to accomplish this is to do it in the context of sum,
42
difference, product, and quotient. The definitions of these functionoperations follow.
Let /and g be functions with a common domain. Then,
(f
and
(fe) (*) = [/(*)][*(*)]
8
Two functions can also be combined to form a third functionusing a process called the composition of functions. If / and g arefunctions, then the composition of these functions is denoted by / « gand is defined by
The domain of this function is defined to be the set of all x, such thatg(x) is in the domain of /. Composition of functions are often illus-trated with pictures. The following picture represents /« g for givenfunctions / and g.
Figure 5.3Here are some examples which show how composition and
computation can be performed.
43
EXAMPLEGiven f(jc) = 2x - 1 and g(*) = Jt + 3 find
(a) (fe)(x).
<fe)* = VW\ fetol
(b)
(c)
+ 3)= 2(^
= 2x +
= 2x + 2
From (b) and (c) above, it is obvious that, in general,
(/"• I?) W *(£'/)(*)•
5.3 INVERSE FUNCTIONSConsider the functions
flx) = 2x and g(x) = 1/2x.
Then / is the "doubling function" and g(x) is the "halving function,"and these functions undo each other. More specifically,
44
and
= 6 = 6
Also, (/= g) (x) = x and (g ./) (x) = x for each real number*. In such acase, / and g are inverses of each other. Here is the general definitionof inverse functions
Two functions / and g are said to be inverses of each other pro-vided that (f . g) (x) = x for each x in the domain of g and(g »/) (x) = x for each x in the domain of/.
The symbol f~\x) is used to represent the inverse of fix). Here is asummary of a procedure for finding /•'(*)•
(1) Lety=y(jc).
(2) Interchange x and y in y = /(*).
(3) Solve the resulting equation for y.
Here is an example which illustrates this procedure.
EXAMPLEGiven f(x) = 2x - 3, find f~\x) and show that these two func-
tions are actually inverse of each other.
+ 3 = 2yx + 3—
-'(„). £±!
oo = /(TO) (f1 •/> w =
45
2 /= (2x -3)
2
= x = X
5.4 INCREASING AND DECREASING FUNCTIONSOn intervals where the graph moves upward from left to right,
the function is said to be increasing. On intervals where the graphmoves downward from left to right, the function is said to be de-creasing. The turning points are those points where the functionchanges from increasing to decreasing or from decreasing to increas-ing. There are two kinds of turning points: relative maxima (highpoints) and relative minima (low points). Examine the graph below.
(*3'/3)
Figure 5.4Using the vertical line test, it is easy to see that this is a graph of afunction. It is an increasing function on the interval where x <: xi andon the interval where x2 < x < x3, and it is a decreasing function onthe interval where xl<x<x2 and where x > x3. The relative maximumpoints are at (xlt y^ and (x3, y3), while the only relative minimum pointis at (x2, y2).
5.5 POLYNOMIAL FUNCTIONS AND THEIR GRAPHSA polynomial in x is an expression of the form
46
where a1; a2, ... and an are real numbers and where all the exponentsare positive integers. When an * 0, this polynomial is said to be ofdegree n. It is common to let P(x) represent
a^c" + an _ iX" ~ ] + ... + a}x + aQ.
Then y = P(x) is a polynomial function. A function with the propertythat
is an even function, while a function with the property
P(-x) = -P(x)
is an odd function.
It would be possible to obtain the graph of a polynomial functiony = P(x) by simply setting up a table and plotting a large number ofpoints; this is how a computer or a graphing calculator operates.However, it is often desirable to have some basic information aboutthe graph prior to plotting points. The graph of the polynomial func-tion, y = a0 is a line which is parallel to the x-axis and | a0 \ units aboveor below the *-axis, depending on whether a0 is positive or negative.A function of this type is called a constant function. The graph of thepolynomial function
y = ^x + a0
is a line with slope fl j and with a0 as the _y-intercept. The graph of thepolynomial function
y = a^x1 + «i* + a0
is a parabola.
It is much more difficult to graph a polynomial function withdegree greater than two. However, here are three items which shouldbe investigated.
(1) Find lines (x-axis and _y-axis) of symmetry and find out whetherthe origin is a point of symmetry. (See Section 2.4.)
(2) Find out about intercepts. The ^-intercept is easy to find but the
47
*- intercepts are usually much more difficult to identify. If pos-sible factor P(x).
(3) Find out what happens to P(x) when | x\ is large.
This procedure is illustrated in the following example.
EXAMPLEGraph
y = x4 - 5x2 + 4
(1) The graph has symmetry with respect to the y-axis.
(2) The y-intercept is at 4. Since
the ^-intercepts are at 1, 2, - 1, and - 2.
(3) As | * | gets large, P(x) gets large.
Here is a sketch of the graph.
J : uFigure 5.5
48
5.6 RATIONAL FUNCTIONS AND THEIR GRAPHSWhen P(x) and Q(x) are polynomials,
P(x)y--^Q(x)is called a rational function. The domain of this function is the set ofall real numbers x with the property that Q(x) * 0.
Graphing rational functions is rather difficult. As is the casefor polynomial functions, it is desirable to have a general procedurefor graphing rational functions. Here is the suggested method for
where P(x) =
Q(x) =+ ... + 0 and
b0.(1) Find lines (*-axis and y-axis) of symmetry and determine
whether the origin is a point of symmetry. (See Section 2.4.)
(2) Find out about intercepts. The y-intercept is at -^ and the jc-in-bo
tercepts will be at values of x where P(x) = 0.
(3) Find vertical asymptotes. A line x = c is a vertical asymptotewhenever Q(c) = 0.
(4) Find horizontal asymptotes
(a) If m = n, then y - — "- is the horizontal asymptote.bm
(b) If m > n, then y = 0 is the horizontal asymptote.
(c) If m < n, then there is no horizontal asymptote.
This procedure is illustrated in the following example.
49
EXAMPLEGraph
y = ~,(1) The axes are not lines of symmetry, nor is the origin a point of
symmetry.
(2) The ^-intercept and the ^-intercept are both at the origin.
(3) The lines x = 1 and x = - 3 are both vertical asymptotes.
(4) The line y = 0 is the horizontal asymptote.
Here is a sketch of the graph.
Figure 5.6
5.7 SPECIAL FUNCTIONS AND THEIR GRAPHSIt is possible to define a function by using different rules for
different portions of the domain. The graphs of such functions aredetermined by graphing the different portions separately. Here is anexample.
50
EXAMPLEGraph
/(*)•x if xsl2* if x>l
Figure 5.7Notice that point (1,1) is part of the graph, but (1, 2) is not.
Functions which involve absolute value can often be completedby translating them to a two-rule form. Consider this example.
EXAMPLEGraph
Sincex if A: a 0
-x if x < 0 ,/(*)
can be translated to the following form.(x-l if xzO{-jc-l if *<0
(See Figure 5.8.)
51
Figure 5.8The greatest integer function, denoted by fix) = [x], is defined by fix) =j, where/ is the integer with the property that; & x <j +1. The graphof this function follows.
Figure 5.9
52
CHAPTER 6
TRIGONOMETRY
6.1 TRIGONOMETRIC FUNCTIONSAn angle is said to be in standard position when its initial side
corresponds to the positive x-axis. In the picture below, a is in stan-dard position and (x, y) denotes a point on the terminal side of a.
a
Figure 6.1Also, r is used to denote the distance between the origin and thepoint corresponding to (x, y), and
The six trigonometric functions are named sine, cosine, tangent, co-tangent, secant, and cosecant and are denoted by sin, cos, tan, cot,sec, and esc. These functions are defined as follows:
53
sin a =
cos a '
tana ••
rx
Xcot a = — ,
sec a = — , x * (Jx
csc a = —•, y x 0yFrom properties of similar triangles, it is easy to show that it does notmake any difference which point is selected on the terminal side ofa. An angle is said to be in a particular quadrant when the terminalside of the angle falls in that quadrant. Here is a table indicatingwhether the trigonometric functions are positive (+) or negative (-)in particular quadrants.
Quadrant for a sin a cos a tan a cot a sec a csc a
II
III
IV
Table 6.1From the definition of the functions, it is obvious that
sec a 1cos a
csc a 1sin a
- , and cot a •• 1tana
This is valuable information when calculators are used in the contextof trigonometry. Most scientific calculators have only keys involvingsin, cos, and tan functions. However, functional values of the secant,cosecant, and cotangent functions can be found with a calculator byfinding the corresponding functional values of reciprocals and usingthe Vj key.
When calculators are used to find functional values of trigono-metric functions, only approximations appear on the display of the
54
calculator. However, there are some angles for which geometricproperties can be used to find the exact functional values. A few ofthese are summarized in the table below.
a
0°
30°
45°
60°
90°
180°
270°
sin a
0
12
\I2
\32
1
0
-1
COS a
1
\ 32
\~22
12
0
-1
0
tana
0
\33
1
V3
undefined
-0
undefined
COt a
undefined
\3
1
\ 33
0
undefined
0
sec a
1
2 \ 33
\"2
2
undefined
-1
undefined
CSC a
undefined
2
\2
2 \33
1
undefined
-1
Table 6.2This table is given with angles measured in degrees. Since 360° = 2itradians, it is easy to translate the table into corresponding radianmeasure categories.
The trigonometric functions are periodic. The graph of eachfunction is shown below.
y = sin x
Figure 6.4
-1
y = sec x
Figure 6.5
56
-ff2
-Jt
-1
y «cscx
0 E2
Figure 6.6y * calx
Figure 6.7Graphs of equations of the form
y = sin (x + a)
are also important. To illustrate the effect of replacing "x" by "x + a,"the graphs of
y = sin x and y = sin (x + n/2)
57
are given. Notice that the graph of y = sin (x + */2) is a "copy" of ysin x which has been moved n/2 units to the left.
Figure 6.8
The graphs of y = sin x and y = 2 sin x are pictured below. No-tice that the amplitude for y = sin x is 1 while the amplitude for y =2 sin* is 2.
Figure 6.9
58
The graphs of y = sin x and y = sin 2x are pictured below. No-tice that y = sin x is periodic with a period of 2rc and y = sin 2x is pe-riodic with a period of JT.
= sin x
Figure 6.10
6.2 INVERSE TRIGONOMETRIC FUNCTIONSA function has an inverse if and only if it is one-to-one. Since
the trigonometric functions are periodic and, thus, not one-to-one,inverses can be established for trigonometric functions only when thedomains are restricted. Here are the definitions of the inverse func-tions for the sine, cosine, and tangent functions.
y = Arcsin x if and only if sin y = x and -n/2sy s n/2
y = Arccos x if and only if cos y = x and 0 sy STC
y = Arctan x if and only if tan y = x and -n/2<x< n!2
6.3 TRIGONOMETRIC IDENTITIESHere is a list of trigonometric identities which are often useful.
esc a = 1
sec a =
sin a
1cos a
59
cot a = 1tana
sin2 a + cos2 a = 1
tan2 a + 1 = sec2 a
cot2 a + 1 = esc2 a
sin (- a) = - sin a
cos (- a) = cos a
tan (- a) = - tan a
sin (a + (3) = sin a cos P + cos a sin (3
cos (a + p) = cos a cos P - sin a sin p
. „, tan a + tan Ptan (a + P) = ——————*-1 - tan a tan P
sin 2a = 2sin a cos a
cos 2a = cos2 a - sin2 a
2 tan atan 2a =
sin" a =
cos2 a '
1 - tan2 al-cos2a
2
l + cos2a
6.4 TRIANGLE TRIGONOMETRYTrigonometry can be applied rather easily to right triangles.
Suppose that right triangle ABC is placed in a coordinate axis position,so that a is in the standard position. (See Figure 6.11.)
Then, from the definition of the sine, cosine, and tangent func-tions,
60
Hypotenuse
Side opposite a
Side adjacent to a
Figure 6.11
sin a =
xcosa = —r
tan a =
but x is the length of the side adjacent to a, y is the length of the sideopposite a, and r is the length of the hypotenuse so
length of side opposite asin a = —fi——————°-———,length of hypotenuse
length of side adjacent to acosa = —e——————-1—————,length of hypotenuse
length of side opposite atan a = ——c——————"-————.length of side adjacent to a
These generalizations are useful when it is necessary to find mea-sures of certain parts of right triangles given measures of other partsof the triangle (solving a triangle).
Here are two generalizations which can be applied in triangle
61
solving situations, when the triangle in question is not necessarily aright triangle.
If a, P, and y are the measures of angles of a triangle and a, b,and c are the lengths of the sides opposite a, (3, and y respec-tively, then,
sin a sina b
sinyc
and
c2 = a2 + b2 - 2ab cos y.
The first of these generalizations is called the law of sines, and thesecond is called the law of cosines.
Determining the area of a triangle given (1) the length of twosides and the measure of the included angle, (2) the measures of twoangles and the length of the included side, and (3) the length of threesides, can be accomplished by the use of the three formulas below.
If a, p, and y are the measures of the angles of a triangle, anda, b, and c are the lengths of the sides opposite a, (3, and yrespectively, and 5 = }/2 (a + b + c), then, the area (K) of thetriangle is
K = — aft sin y
„ 1 2 sin(3sinyK = -a — - —— - and2 sin a
6.5 SOLVING TRIGONOMETRIC EQUATIONSThe procedures for solving trigonometric equations are similar
to the procedures for solving algebraic equations. However, it isoften desirable to use substitutions, so that the equation being solvedcontains only one of the trigonometric functions. The example belowillustrates this technique.
62
EXAMPLESolve
sin x + 1 = 2 cos2* for 0 s x < 360°.
sin x + 1 = 2(1 - sin2*)
sin x + 1 = 2 - 2 sin2*
2 sin2* + sin * - 1 = 0
(2s in*- l ) (s in*+ 1) = 0
2 sin x - 1 = 0 or sin x + 1 = 0
sin * = ' /2 or sin* = — 1
* = 30°, 150°, 270°.
6.6 THE TRIGONOMETRIC FORM OF A COMPLEXNUMBERIn an earlier section, it was determined that when a and b are real
numbers, a + bi is a complex number. Then, it is natural to associatea + bi with the point (a, b). It is obvious that this procedure establishesa one-to-one correspondence between the set of all complex numbersand the set of all points in a plane.
(a,b)
Figure 6.12A segment is drawn from the origin to (a, b). Then, 6 is the angle
63
formed by this segment and the positive x axis, and r is the distancefrom the origin to (a, b).lfz = a + bi, then
Z = r(cos 0 + / sin 9)
and r(cos 0 + / sin 9) is called the trigonometric form of Z.
Products and quotients of two complex numbers can be deter-mined by the formulas below.
If Z] = ri(cos 9! + / sin 9j) and Z2 = r2(cos 92 + / sin 92)
then ZtZ2 = rir2[cos(0, + 92) + i sin (9, + 92)]
and —L--5-[cos(61-e2) + Jsin(01-e2)] r2 * 0Z2 r2
The product rule extends to n factors and when the n factors areidentical, and if Z = r(cos 9 + i sin 9), then
Z" = r" (cos «0 + i sin n9).
This is called De Moivre's Theorem.
Suppose Z = r(cos 9 + i sin 9). Then applying De Moivre'sTheorem to
M . .rn\ cos — + isin —n n
'"{cos —\ n
0 . . 9JSHI —n
It follows thati
r-( 0" cos —\ n
0 .
9 . . 9cos— At + i s m — nn n
• r(cos9 + z'sin9)Z
64
is an «lh root of Z. Also, since
cos 6 = cos(8 + K • 360°) and sin 6 = sin(9 + K • 360°),
it is possible to find the n, nlh roots of Z. This process is illustrated inthe example below.
EXAMPLE
Find the four fourth roots of 2 + 2\ 3i.
Z = 4(cos 60° + i sin 60°),
Z = 4(cos 420° + / sin 420°)
Z = 4(cos 780° + i sin 780°),
Z = 4(cos 1140° + / s i n 1140°)
60° .. 60W,
470° 420
.T/ 60° . . 60°\ - . .44 cos —— + zsm—— = \ 2 (cos!5 + «sin 15 )V 4 4 /
] = \ I (cos 105°+i sin 105°)
~ / 780° 78fl°\ _^ 4 4 cos-^^- + Jsin-^- =\2(cosl95° + /sinl95°)
V 4 4 /
cos^^- + isin^^) - \ 2 (cos285°+ /sin 285°)
65
CHAPTER 7
EXPONENTS AND LOGARITHMS
7.1 INTEGER EXPONENTSThis section contains a summary of important interpretations
of exponents. Here is a definition which describes the meaning ofinteger exponents.
If n is a positive integer, and if x is a real number, then
x" = x-x-x-x-x...xn factors
x° = 1 and
~From these definitions, the following exponent generalizations canbe derived.
If a and b are real numbers, and if m and n are integers, then
am-an=am+n
m
„"
66
(abf = a"b"
a}" = a^b) b"
7.2 RATIONAL NUMBER EXPONENTSRadicals were discussed in Section 3.4 and since radicals involve
rational number exponents,. a short review may be desirable. Here isa useful definition.
Suppose that m and n are positive integers with n > 1, and sup-pose that a is a real number such that 1 a is a real number. Then
m~a"
Of course '('a is not a real number when n is even and a is negative.
With this definition, it is easy to see that the exponent lawslisted in Section 7.1 can be extended from integers to positive rational
ffi _numbers. Then it is quite easy to see that a" = 1 am . Also,
and this describes what happens when the exponent is a negativerational number. Of course, the exponent laws described in Section7.1 carry over to rational number exponents.
7.3 EXPONENTIAL FUNCTIONSThe previous section includes an interpretation of rational
number exponents. Now, a pertinent question to ask is, "What hap-pens when the exponent is an irrational number?" Since V2 « 1.4142,
67
1.41
1.414 1-415
Using this process, it can be shown that 2 x 2 is in fact a single realnumber. This idea can be generalized and if b is a real number, b > 0,b* is a real number for every real number x. Then
is called an exponential function. Here is the graph of fix) = 2".
-3 -2 -1
Figure 7.1The irrational number e is defined in the following way.
1 \ "lim(l + — j =en~»oo\ nJ
It can be shown that e - 2.71828.
7.4 LOGARITHMSA logarithm is an exponent. This is illustrated in the following
definition.
For b > 0, b * 1 and for x> 0,
68
y = if and only if by = x.
Here are some examples which illustrate the use of this definition.
EXAMPLEFind Iog28.
23 = 8
Thus, Iog28 = 3.
Find Iog2 V2-
Thus, log2V2 =-1.
Find Iog2 \ 2 .
21/2 = V2
Thus, Iog2 v"2 = V2.
Find Iog2 (- 2).
Since there does not exist a real number x such that
2X = - 2, Iog2(- 2) is undefined.
Since a logarithm is an exponent, it is easy to use exponent laws toestablish the following generalizations.
If Xi, x2, and b are positive real numbers, with b * 1, and mrepresents any real number,
logb x"1 = m
= m.
and
69
The irrational number e was defined in Section 7.3. This num-ber is often used as a base in the investigation of logarithms. Aspecial symbology "In" has been developed for this situation. Spe-cifically,
log^a = In a.
The properties of exponents can be used in equation solvingsituations. Here are some examples.
EXAMPLESolve
In 5 = In 9 + 1 - 2x
2x = 1 + In 9 - In 5
l + ln9-ln5x = ———————2
Solve
logjx + Iog3(.x + 2) = 1logjx(* + 2) = 1
x(x + 2) = 31
x2 + 2x - 3 = 0
x = - 3 or x = 1
However, x cannot be a negative number. Thus, the only solu-tion is x = I .
70
Solve
2* = 5
Iog2 2* = Iog25
x = Iog2 5
It is sometimes desirable to change from one logarithmic baseto another. The following formula illustrates this process.
7.5 LOGARITHMIC FUNCTIONSA function of the form
is called a logarithmic function. In Section 7.4, it was pointed outthat logi* is meaningful only when b > 0 and b * I . Here are someproperties of f(x) =
(1) The domain of /is the set of all positive real numbers.
(2) The range of /is the set of all real numbers.
(3) y(i) = o(4) / is an increasing function when b > 1 and decreasing if 0 < b
<1.
From the definition of a logarithm, it is obvious that
are inverse functions. The graph of
fix) = 2- and f(x) =
are pictured below.
71
> = 2"
/(jr) = logjjf
Figure 7.2
72
CHAPTER 8
CONIC SECTIONS
8.1 INTRODUCTION TO CONIC SECTIONSConic sections are the curves formed when a plane intersects
the surface of a right circular cone. As shown below, these curves arethe circle, the ellipse, the parabola, and the hyperbola.
8.2 THE CIRCLEA circle is defined to be the set of all points at a given distance
from a given point. The given distance is called the radius, and thegiven point is called the center of the circle. Using the distanceformula described in Section 2.3, it is easy to establish that an equa-tion of a circle with center of (h, k) and radius of r is
(x - h}2 + (y - = r2
An equation of this kind is said to be in standard form. Here are twoexamples which illustrate the practicality of the standard form.
EXAMPLEFind an equation of the circle with center of (2, - 3) with
radius of 6.
(x - 2)2 + (y + 3)2 = 36
Graph the equation listed below. (See Figure 8.2.)
x2 + y2-6x+ 10y + 30 = 0
(x2 - 6x) + (y2 + 10>0 = - 30
Figure 8.2
74
(x2 - 6x + 9) + (y2 + Wy + 25) = - 30 + 34
(x - 3)2 + 0 + 5)2 = 4
Notice, in the second example, how easy it is to graph
(x - 3)2 + (y + 5)2 = 4
and how difficult it would be to graph
x2+y2-6x + 10y + 30 = 0
in that form.
8.3 THE ELLIPSEAn ellipse is defined to be the set of all points in a plane, the
sum of whose distances from two fixed points is a constant. Each ofthe fixed points is called a focus. The plural of the word focus is foci.A simple method of constructing an ellipse comes directly from thisdefinition. Mark two of the foci and call them Fl and F2, then insert athumb tack at each focus. Next, take a string which is longer than thedistance between F, and F2 and tie one end at Fj and the other end atF2. Then, pull the string taut with a pencil and trace the ellipse. Itwill be oval shaped and will have two lines of symmetry and onepoint of symmetry.
Figure 8.3Using the distance formula, it is fairly easy to derive an equation ofthe ellipse with foci at (c, 0) and (- c, 0) with the sum of the distances
75
of la. The standard equation of such an ellipse is2 2
where b2 - a2 - c2. Obviously, a2 > b2. Here is a graph of such an el-lipse.
Figure 8.4The center of the ellipse is the point midway between the foci.In this case, the center is at the origin. The segment from (- a, 0) to(a, 0) is called the major axis and its length is 2a. The segment from(0, - b) to (0, b) is called the minor axis and its length is 26. In the el-lipse, the x-axis and the y-axis are lines of symmetry and the origin isa point of symmetry. The eccentricity is defined to be c/fl. When thisratio is close to 0, the ellipse resembles a circle, but when this ratio isclose to 1, the ellipse is elongated.
It is very easy to graph an ellipse when the standard equation isgiven. In most cases, it is desirable to convert to that form. Thisprocess is illustrated in the example below.
EXAMPLEGraph
9x2 + 16y2 = 144
76
144 144144144
162
Figure 8.5
For an ellipse with foci at (0, c) and (0, - c), and with the sum of dis-tances as 2a, the standard equation is
v2 x2
*2+*2-la2 b2
Here is a table which illustrates what happens when the centerof the ellipse is not at the origin.
Center of Sum ofellipse distances Foci Standard equation
(h,k) 2a
(h,k) 2a (h,k+c)&nd(h,k-c)
In both cases, b2 = a2 - c2.
8.4 THE PARABOLAA parabola is defined to be the set of all points in a plane
equally distant from a fixed line and a fixed point not on the line.The line is called the directrix and the point is called the focus. Theaxis of a parabola is the line which passes through the focus and
77
which is perpendicular to the directrix. The vertex of the parabola isat the point which is half way between the focus and the directrix. Apicture of a parabola is shown in Figure 8.6.
It is somewhat common for the undirected distance betweenthe directrix and the focus to be labelled 2c. (This means that c > 0.)Here is a table which relates the standard equation to the position ofthe vertex, the position of the focus, and the equation of the directrix.
DirectionDirectrix
y = -c
X = -C
y = c
X = C
y = -c+k
x = -c+h
y = c+k
x = c+h
Focus
(0,c)
(c,0)
(0,-c)
(-c,0)
(h, c+k)
(c+h, k)
(h, -c+k)
(-c+h, k)
Vertex
(0,0)
(0,0)
(0,0)
(0,0)
(h,k)(h,k)
(h,k)
(h,k)
Standard equation Parabola Opens
x2 - 4cy
y2 = 4cx
x2 = -4cx
y2 = -4cx
(x-h)2 = 4c(y-k)
(y-k)2 = 4c(x-h)
(x-k)2 = - 4c(y-h)
(y-k)2 = -4c(x-h)
Up
Right
Down
Left
Up
Right
Down
Left
The table above illustrates that once an equation for a parabola is instandard form, it is easy to characterize the parabola. Consider thefollowing.
y2 - 6y - 6x + 39 = 0
y2 - 6y = 6x - 39
y2 - 6y + 9 = 6x - 30
This is a parabola with vertex at (5, 3) and focus at (6'/2, 3). Theequation of the directrix is x = 3l!2 and the parabola opens to the right.
78
vertex
Figure 8.6In general, equations of the form
y2 + Dx + Ey + F = 0
are parabolas which open to the right or left, and equations of theform
x2 + Dx + Ey + F = 0
are parabolas which open up or down.
8.5 THE HYPERBOLAThe hyperbola is defined to be the set of all points in a plane,
the difference of whose distances from two fixed points is a constant.If the foci are at (c, 0) and (- c, 0), and if the difference of the dis-tances is 2a with 0 < a < c, then the standard equation for this hyper-bola is
x2
a2yLb2 -i,
where b2 = c2 - a2. A graph of the hyperbola follows.
79
Figure 8.7Notice that the hyperbola has two branches.
For the hyperbola, the focal axis is the line through the foci,the center is the point midway between the foci, and the conjugateaxis is the line through the center which is perpendicular to the focalaxis. The hyperbola has symmetry with respect to those lines andthat point. For the hyperbola pictured above, the focal axis is the ;c-axis,the conjugate axis is the _y-axis, and the center is at the origin. As-ymptotes are described in Section 2.5. Notice that, in this case, thelines
b , -by = — x and y = — xa a
are asymptotes for the hyperbola pictured.
If the foci are at (0, c) and (0, - c), and if the difference betweenthe distances is still 2a with 0 < a < c, then the standard equation forthe hyperbola is
v2 x22— __ -ia2 f c 2 " 1 '
where b2 = c2 - a2. In this case, the center is at the origin, the focal
80
axis is the _y-axis, the conjugate axis is the ;t-axis, and the hyperbola hasthe lines
a —ay - — x and y = — xb y b
as asymptotes. A graph of this hyperbola follows.
Figure 8.8The standard form of the equation of the hyperbola with center
at (h, k) and with focal axis parallel to the x-axis is
'1,
and the standard form for the equation of the hyperbola with centerat (h, k) and with focal axis parallel to the y-axis is
(v-A:)2 (x-h)2= 1.
81
CHAPTER 9
MATRICES AND DETERMINANTS
9.1 MATRICES AND MATRIX OPERATIONSA matrix is a rectangular array of numbers, usually real numbers.
These numbers are called entries. Matrices are classified according totheir number of rows and columns. An m by n matrix has m rows andn columns and such a matrix is said to have size m by n. Two matri-ces of the same size are equal when the corresponding entries areequal,
To add matrices of the same size, merely add the correspondingentries. The following is a description of properties concerning theaddition of matrices.
Suppose A, B, and C are arbitrary m by n matrices, O is the mby « matrix with 0 as each entry, and -A is the m by n matrix ineach entry in - A is the opposite of the corresponding entry inA. Then
A + B = B +A
A +O=O+A = A
To subtract two matrices of the same size, merely subtract the
82
corresponding entries. To multiply a matrix by a number, multiplyeach entry in the matrix by that number. This is called scalar multi-plication and the numbers (usually real numbers) are called scalars.The following is a significant scalar multiplication property.
If A is an m by « matrix and a and b are scalars, then
(ab)A =a(M)
The standard product of two matrices is much more difficult tofind than the sum or difference of two matrices, or the scalar productof a number and a matrix. If A is an m by p matrix and B is a p by nmatrix, then AB is an m by n matrix where the entry in the ith row and7th column ofAB is the sum of all the products found from multiplyingentries in the zth row of A with the/h column of B. Note, for the prod-uct AB to exist, A must have the same number of columns as B hasrows. Of course, this means that when A and B are of the same size,the product may not even exist. For that reason, it is somewhatcommon to primarily confine matrix multiplication to square matrices.Here are two properties concerning products of square matrices.
Suppose A, B, and C are arbitrary n by « matrices and / is the nby n matrix with entries in the first row first column, secondrow second column, third row third column, ... nth row nih
column all 1, and with all other entries 0. Then
(AB)C=A(BC)
It is somewhat obvious that there are n by n matrices .A and B, such thatAB * BA. In many cases, there exists an n by n matrix A~l with theproperty that A A~l = A~: A = I. Section 9.3 is devoted to determiningwhen square matrices have multiplicative inverses and how to findthem when they do exist.
9.2 DETERMINANTSAssociated with each n by n matrix A, there is a specific real
number called the determinant of A and sometimes denoted by 6(A).The usual way to calculate the determinant for a 2 by 2 matrix is
83
described below.
If
A- \an flna22
It is common to use vertical bars to represent determinants.Then
504) =an ana2\ a22
-,,•*"»*,.
Here are three important definitions concerning the calculation ofdeterminants for a matrix A where
«22 «23 2n
n\
(1) The minor, A/,y, of atj is the determinant of the (n - 1) by (n - 1)matrix obtained by deleting the ilh row and jlh column of A.
(2) The cofactor, A^, of a,y is
(3) The determinant of A is the sum of the n products formed bymultiplying each entry in a single row (or column) by its cofac-tor.
Here is an example which illustrates this procedure.
EXAMPLE0 1 11 1 02 -4 1
-«-.y 1 1-4 1 *.(-,>• 0 1
2 1 ^-.y 0 12 -4
0(-2)••-1
84
In this instance, the determinant was determined by expanding aboutthe second row. Of course, the determinant could be determined byexpanding about any row or column.
Determinants have some properties that are useful because theysimplify the calculations. Here are these properties.
(1) If each element in any row or each element in any column is 0,then the determinant is equal to 0.
(2) If any two rows or any two columns of a determinant areinterchanged, then the resulting determinant is the additive in-verse of the original determinant.
(3) If two rows (or columns) in a determinant are equal, then thedeterminant is equal to 0.
(4) If each entry of one row (or column) is multiplied by a realnumber c and added to the corresponding entry in another row(or column) in the determinant, then the resulting determinantis equal to the original determinant.
(5) If each entry in a row (or column) of a determinant is multipliedby a real number c, then the determinant is multiplied by c.
Here is an example which illustrates using these last two generaliza-tions.
EXAMPLE
15 1 1318 1 16 =321 -1 21
= 3
= 3
5 1 136 1 167 -1 21
5 + 7 ! + (-!) 13 + 216 + 7 ! + (-!) 16 + 217 -1
12 0 3413 0 377 -1 21
21
85
= 3 0(-1)3 13 377 21
0 + (12-37-13-34)]
12 347 21
12 3413 37
3-26
9.3 THE INVERSE OF A SQUARE MATRIXFor a given n by n matrix A, if there is an n by n matrix A~'
with the property that A A'1 = A^ A = /, then A*1 is the multiplicativeinverse of A. This section is devoted to determining when a squarematrix has an inverse and the procedures for finding inverses, if theyexist. Consider the square matrix A where
«12
«22"l/i
If b(A) is the determinant for A and 5(/\)and
0, then A has an inverse A"
-1 1~ £>(A)
f\\ 1 ./ITI . . • ./I-,!11 Z.1 n 1
A\2 A22 ... An2
In In • ' ' nn
,
where Aig is the cofactor of aig. If 6(A) = 0, then A has no inverse.While this generalization describes a definite process for determininginverses, the process is very laborious for large matrices and, thus, isnot very practical.
A procedure for determining inverses which is much less de-manding, from a computational standpoint, makes use of what isknown as elementary row operations. Here are the elementary rowoperations for an m by n matrix A.
(1) Multiply each row of A by a nonzero real number c.
86
(2) Interchange any two rows of A.
(3) Multiply the entries of any row of A by a real number c and addto the corresponding entries of any other row.
If B is a matrix resulting from a finite number of elementary rowoperations on A, then A and B are row-equivalent matrices, and thisis denoted by A ~ B. If the inverse of an n by n matrix A exists, then itcan be obtained by using elementary row operations. The followingresult illustrates how this can be done.
If A has an inverse and if [A \ I ] is the n by 2n matrix obtainedby adjoining the n by n identity matrix ioA, then
Here is an example which illustrates how this theorem can be ap-plied.
Given
A =5 0 22 2 1-3 1 -1
using elementary row operations.
5 0 22 2 1-3 1 -1
1 0 0 '0 1 00 0 1
[5 + (-2)2 0 + (-2)2 2 + (-2)1 12 2 1-3 1
1 - 4 02 2 1-3 1 -1
~
1 -2 00 1 00 0 1
-1
1 -4 02 + (-2)1 2 + (-2)(-4) l + (-2)(-3 + 3(1) l + 3(-4) -1 + 3(0
+ (-2)0 0 + (-0 10 0
1) 0 + (-2)l 1 +) 0 + 3(1) 0
0 + (-201
-2
0 + 3(-2)
00 + (-2)01 + 3(0)
87
1 -4 00 10 10 -11 -1
~
1 -2 0- 2 5 03 -6 1
1 - 4 0 1 - 2 0 '0 + 0 10 + (-11) ! + (-!) -2 + 3 5 + (-6) 0 + 1
0 - 1 1 - 1 3 - 6 1
1 -4 00 - 1 00 -11 -1
~
~
113
1 -4 00 1 00 -11 -1
-2 0 '-1 1-6 1
1 - 2 0 "-1 1 -13 -6 1
1 + 4(0) -4 + 4(1) 0 + 4(0) l + 4(-l) -2 + 4(1) 0 + 4(-l)"0 1 0 - 1 1 - 1
0 + 11(0) -11 + 11(1) -1 + 11(0) 3 + ll(-l) -6 + 11(1) 1 + 11(-1)
1 0 0 - 30 1 0 - 10 0 - 1 - 8
~1 0 00 1 00 0 1
-3-18
2 -41 -15 -10
2 -41 -1
-5 10
Thus,-3 2 -4-1 1 -18 -5 10
88
CHAPTER 10
MISCELLANEOUS TOPICS
10.1 THE FUNDAMENTAL COUNTING PRINCIPLE,PERMUTATIONS, AND COMBINATIONSIf a first task can be performed in m ways, and if after the first
task has been performed a second task can be performed in n ways,then the two tasks can be performed in mn ways. This is called thefundamental counting principle.
A permutation of r objects from a set of n objects is an orderedarrangement of those r objects. The symbol ,Pr is used to represent thenumber of permutations of n objects taken r at a time, and when w a r .
n\(n-r)l
A combination of r objects from a set of n objects is an unor-dered selection of r objects from the n objects. The symbols (*) andnCr are both used to represent the number of combinations of nobjects taken r at a time and
n~-0 n!r!(n-r)!
89
10.2 THE BINOMIAL THEOREMThe Binomial Theorem involves the expansion of products of
the form (a + b)n, where n is a positive integer. When n and k are
positive integers with k s n, the symbol , represents the number of\*/
combinations of n different things taken k at a time, and as indicatedin the previous section,
n\k\(n-k)\
The Binomial Theorem is as follows:
The example below illustrates the use of this famous theorem.
EXAMPLE(2x-3y)4 = ( \ (2x)4 (2xf(-3yf
: 16*2 -96x3y + 216*2y2 - 2l6xy3 + Sly4
10.3 SEQUENCES AND SERIESA sequence is a function whose domain is the set of all natural
numbers. All the sequences described in this section will have asubset in the set of all real numbers as their range. It is common tolet an represent the «th term of the sequence. For example, if
an = lOn
then the sequence is 10, 20, 30, 40 ....
The sum of the first n terms of the sequence,
90
«!, a2,
is indicated by
a, +« 2 + a 3 + ... +an,
and the sum of these terms is called a series. The Greek letter I, is usedto represent this sum as indicated below.
n^ ak = a ] + a 2 + a 3 + ... + «„k-l
For a fixed number a and a fixed number d, the sequence
a, a + d, a + Id, a + 3d + ...
is called an arithmetic sequence, and the n* term of this sequence isgiven by
an = a + (n-\)d
The number a is called the first term and d is called the common dif-ference. The symbol Sn is used to represent the corresponding seriesand
and
A sequence of the form
a, ar, ar2, ...
is called a geometric sequence, where a is the first term and r is thecommon ratio. The nlh term of such a sequence is
an = ar"-1
The symbol Sn is used to represent the corresponding series and
91
Expressions of the form
a + ar + ar2 ...
are called infinite geometric series. When | r \ < 1, the sum S of theinfinite geometric series exists and
5-- "1-r
10.4 MATHEMATICAL INDUCTIONThe material in this section refers to the set of all natural
numbers N. Suppose A is a set with the properties
(1) ACN
(2) I G A a n d
(3) i f K E A then/C + 1 EA.
It can be concluded that in this case, A = N.. This generalization pro-vides the basis for proof by mathematical induction. The correspond-ing theorem is as follows:
If for a given statement about positive integers, the statement istrue for 1 and if it is true for n = k, then it is true for n = k + 1,then the statement is true for all positive integers.
Here is an example which illustrates the use of this theorem.
EXAMPLEProve that
1 + 2 + 3 + . . . + / i =
Proof:
Now assume that the statement is true when n = k. Then
92
+ 2)
Thus, the statement is true when n = k + 1. Thus, the statement is truefor all positive integers.
10.5 ZEROS OF A POLYNOMIALAs indicated in Section 3.5, an expression of the form
is a polynomial and a function of the form
P(x) = ajc" + an _ }x" - ] + . . . a,* + aQ
is a polynomial function. A number r is said to be a zero of this poly-nomial provided that P(r) = 0. The Factor Theorem, which follows,illustrates the relationship between zeros and factors of a polynomial.
For a polynomial P(x), r is a zero of P(x) if, and only if, x - r isa factor of P(x). Here are two other theorems, which are useful interms of finding zeros of polynomials.
If P(x) is a polynomial, and if P(a) < 0 and P(b) > 0, then P(x)has a zero between a and b.
If Pjq is a rational number in lowest terms which is a zero ofP(x) where
P_(x) = aj." + a,,.! + ... a}x + a0
and where a0, a\, a2, ••• and an are all integers, then/* is a fac-tor of a0 and q is a factor of an.
93
Here is an example which illustrates the use of these theorems.
EXAMPLEFind the zeros of
fix) = 2x* - 3*3 - lx2 - 8x + 6.
Then the possible rational zeros are
± 1, ±2, ±3, ±6, ±l/2, and ± 3/2. Applying standard divisionprocedures as follows,
2x3 + 3x2 + 2x - 2
x-3 I 2X4- 3*3- lx2- 8x+ 62x*- 6x3
3x3 - lx2
TvS QV2•JA 7A
2x2- 8x2x2- 6x
-2x
-2x
indicate that 3 is a zero of this polynomial and
2X4 - 3*3 - lx2 - 8x + 6 = (x- 3) (2x3 + 3x2 + 2x- 2).
Then
2x2+ 4x+ 4
2x-2- x2
4x2+ 2x4x2- 2x
4x-2
4x-2
94
and Zx4 - 3*3 - 7x2 - 8* + 6 = (x - 3) (x - V2 ) (2x2 + 4x + 4).
Applying the quadratic formula to
2x2 + 4x + 4 = 0
gives - 1 ± i as solutions. Thus, the zeros of
2x* - ax3 - 7x2 - 8* + 6
are 3, l/2 , - 1 + i, and - 1 - i.
Next, consider the polynomial
The only possible rational zeros of this polynomial are ±1 and ± 3.However, none of these numbers are actually zeros of the polynomialwhich means that this polynomial has no rational zeros. However, itis possible to approximate zeros of this polynomial. Since
/(I) = -3 and /(2) = 1,
fix) has a zero between 1 and 2. Also /(1.8) = - 0.408 and /(1. 9) =0.249 so the polynomial has a zero between 1.8 and 1.9. This processcan be continued so the zero could be approximated to any desiredlevel of accuracy.
The procedures described in this section apply only to findingrational number zeros of polynomials. However, when a polynomialhas no rational zeros, it is possible to approximate irrational zerosusing the process described in the example above. A calculator or acomputer is a vital tool in this process.
It is often very difficult to find nonreal number zeros of poly-nomials. Here is one theorem concerning nonreal zeros.
If
are real numbers and if a + bi is a zero of f(x), then a - bi is azero of /(*).
This theorem is applied in the following example.
95
EXAMPLEFind all the zeros of
f(x) = x4 - 2x3 + 3x2 - 2x + 2
given that 1 + i is one zero. Since 1 + t is a zero, 1 - / is a zero andthere is a polynomial Q(x) such that
But since
= (x2 - 2x + 2) Q(x).
2>x2 - 2x + 2 ,+ 1
P(x) = (x2 - 2x + 2) (x2 + 1)
Thus, the zeros of this polynomial are 1 + i, \ - i, i, and - /.
Now consider the polynomial
where fe0> ^i • • • and ^« are complex numbers. Such a polynomial issaid to be a polynomial over the complex numbers. Here is the funda-mental theorem of algebra.
Every polynomial of degree n a 1 over the complex numberhas at least one complex number zero.
The factor theorem stated earlier in this section can be extended topolynomials over the complex numbers. Here is a related theorem.
Every polynomial of degree n a 1 over the complex numberscan be expressed as a product of a constant and linear factorsof the form (x - xt) where Xj is a complex number.
If a factor x - Xj occurs k times in a linear factorization of P(x), then x,is said to be a zero of multiplicity k. This means that every polyno-mial with complex number coefficients has exactly n zeros.
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