Intermediate Algebra Textbook

525 Pages·2015·4.13 MB·English

Inntermmediate AAlgebbra TTextbbookk
for SSkyliine CColleege
Adapted for Skyline College
from text created by
Department of Mathematics
College of the Redwoods
Acknowledgments
Authored and edited by David Arnold, Michael Butler, Mike Haley, Diane Harrow, Aeron
Ives, Stephen Jackson, Craig Kutil, Teresa Matsumoto, Jacob Miles Prystowsky, Todd
Olsen, David Tuttle, and Bruce Wagner.
Design, layout, and technical work by Jacob Miles Prystowsky.
Additional assitance from Aditya Mahajan, Hans Hagen, Taco Hoekwater, and the
ConTEXt mailing list.
Funded by a grant from the Project for Learning Enhancement.

Skyline College Edition Contributors
Department of Mathematics, Skyline College
Cover photo by Tadashi Tsuchida

Changes to the Skyline College Edition:
Chapter 2.5
Chapter 4.1

Copyright
All parts of this intermediate algebra textbook are copyrighted in the name of
Department of Mathematics, College of the Redwoods. They are not in the public
domain. However, they are being made available free for use in educational in-
stitutions. This offer does not extend to any application that is made for profit. Users
who have such applications in mind should contact David Arnold at david-
[email protected] or Bruce Wagner at [email protected].

This work (including all text, Portable Document Format files, and any other orig-inal
works), except where otherwise noted, is licensed under a Creative Commons
Attribution-NonCommercial-ShareAlike 2.5 License, and is copyrighted ©2006,
Department of Mathematics, College of the Redwoods. To view a copy of this license,
visit http://creativecommons.org/licenses/by-nc-sa/2.5/ or send a letter to Creative
Commons, 543 Howard Street, 5th Floor, San Francisco, California, 94105, USA.
Contents
Chapter 1 Factoring ................................................................................................................... 2 
1.1   The Greatest Common Factor ..................................................................................... 2 
1.2  Solving Nonlinear Equations ...................................................................................... 15
1.3  Factoring ax2+bx+c where a=1 .................................................................................. 29
1.4  Factoring ax2+bx+c where a1 .................................................................................. 43
1.5  Factoring Special Forms ............................................................................................ 56 
1.6  Applications of Factoring ............................................................................................ 71
Chapter 2 Quadratic Functions ............................................................................................ 79 
2.1  The Parabola ................................................................................................................ 80 
2.2  Vertex Form ................................................................................................................ 101 
2.3  Zeros of the Quadratic .............................................................................................. 119 
2.4  The Quadratic Formula ............................................................................................. 139 
2.5  Complex Numbers ..................................................................................................... 161 
2.6  Optimization ................................................................................................................ 167
Chapter 3 Rational Functions ............................................................................................. 183 
3.1  Introducing Rational Functions ................................................................................ 184 
3.2  Reducing Rational Functions ................................................................................... 199 
3.3  Products and Quotients of Rational Functions ..................................................... 216 
3.4  Sums and Differences of Rational Functions ........................................................ 233 
3.5  Complex Fractions..................................................................................................... 247 
3.6  Solving Rational Equations ...................................................................................... 262 
3.7  Applications of Rational Functions .......................................................................... 279 
3.8  Direct and Inverse Variation .................................................................................... 295
Chapter 4 Exponential and Logarithmic Functions ...................................................... 304 
4.1  Introduction to Exponential Functions .................................................................... 305 
4.2  Exponents and Roots ................................................................................................ 317 
4.3  Exponential Functions .............................................................................................. 336 
4.4  Applications of Exponential Functions ................................................................... 353 
4.5  Inverse Functions ...................................................................................................... 367 
4.6  Logarithmic Functions ............................................................................................... 385 
4.7  Properties of Logarithms .......................................................................................... 395 
4.8  Exponential Growth and Decay ............................................................................... 406
Chapter 5 Radical Functions............................................................................................... 419 
5.1.  The Square Root Function ....................................................................................... 420 
5.2  Multiplication Properties or Radicals ...................................................................... 435 
5.3  Division Properties of Radicals ................................................................................ 455 
5.4  Radical Expressions .................................................................................................. 472 
5.5  Radical Equations ...................................................................................................... 491 
5.6  The Pythagorean Theorem ...................................................................................... 505
Index .......................................................................................................................................... 526
F
(cid:36)(cid:73)(cid:66)(cid:81)(cid:85)(cid:70)(cid:83) (cid:18) (cid:66)(cid:68)(cid:85)(cid:80)(cid:83)(cid:74)(cid:79)(cid:72)
(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)
(cid:53)(cid:73)(cid:70)(cid:1)(cid:66)(cid:79)(cid:68)(cid:74)(cid:70)(cid:79)(cid:85)(cid:1)(cid:35)(cid:66)(cid:67)(cid:90)(cid:77)(cid:80)(cid:79)(cid:74)(cid:66)(cid:79)(cid:84)(cid:1)(cid:77)(cid:70)(cid:71)(cid:85)(cid:1)(cid:85)(cid:73)(cid:70)(cid:1)(cid:70)(cid:66)(cid:83)(cid:77)(cid:74)(cid:70)(cid:84)(cid:85)(cid:1)(cid:70)(cid:87)(cid:74)(cid:69)(cid:70)(cid:79)(cid:68)(cid:70)(cid:1)(cid:80)(cid:71)(cid:1)(cid:85)(cid:73)(cid:70)(cid:1)(cid:86)(cid:84)(cid:70)(cid:1)(cid:80)(cid:71)(cid:1)(cid:82)(cid:86)(cid:66)(cid:69)(cid:83)(cid:66)(cid:85)(cid:74)(cid:68)(cid:1)(cid:70)(cid:82)(cid:86)(cid:66)(cid:14)(cid:85)(cid:74)(cid:80)(cid:79)(cid:84)(cid:1)(cid:80)(cid:79)(cid:1)(cid:68)(cid:77)(cid:66)(cid:90)(cid:1)
(cid:85)(cid:66)(cid:67)(cid:77)(cid:70)(cid:85)(cid:84)(cid:1)(cid:69)(cid:66)(cid:85)(cid:74)(cid:79)(cid:72)(cid:1)(cid:67)(cid:66)(cid:68)(cid:76)(cid:1)(cid:85)(cid:80)(cid:1)(cid:18)(cid:25)(cid:17)(cid:17)(cid:1)(cid:35)(cid:36)(cid:15)(cid:1)(cid:53)(cid:73)(cid:70)(cid:90)(cid:1)(cid:86)(cid:79)(cid:69)(cid:70)(cid:83)(cid:84)(cid:85)(cid:80)(cid:80)(cid:69)(cid:1)(cid:73)(cid:80)(cid:88)(cid:1)(cid:85)(cid:73)(cid:70)(cid:1)(cid:66)(cid:83)(cid:70)(cid:66)(cid:1)(cid:80)(cid:71)(cid:1)(cid:66)(cid:1)(cid:84)(cid:82)(cid:86)(cid:66)(cid:83)(cid:70)(cid:1)(cid:68)(cid:73)(cid:66)(cid:79)(cid:72)(cid:70)(cid:84)(cid:1)(cid:88)(cid:74)(cid:85)(cid:73)(cid:1)(cid:85)(cid:73)(cid:70)(cid:1)
(cid:77)(cid:70)(cid:79)(cid:72)(cid:85)(cid:73)(cid:1)(cid:80)(cid:71)(cid:1)(cid:74)(cid:85)(cid:84)(cid:1)(cid:84)(cid:74)(cid:69)(cid:70)(cid:15)(cid:1)(cid:39)(cid:80)(cid:83)(cid:1)(cid:70)(cid:89)(cid:66)(cid:78)(cid:81)(cid:77)(cid:70)(cid:13)(cid:1)(cid:85)(cid:73)(cid:70)(cid:90)(cid:1)(cid:76)(cid:79)(cid:70)(cid:88)(cid:1)(cid:74)(cid:85)(cid:1)(cid:88)(cid:66)(cid:84)(cid:1)(cid:81)(cid:80)(cid:84)(cid:84)(cid:74)(cid:67)(cid:77)(cid:70)(cid:1)(cid:85)(cid:80)(cid:1)(cid:84)(cid:85)(cid:80)(cid:83)(cid:70)(cid:1)(cid:79)(cid:74)(cid:79)(cid:70)(cid:1)(cid:85)(cid:74)(cid:78)(cid:70)(cid:84)(cid:1)(cid:78)(cid:80)(cid:83)(cid:70)(cid:1)(cid:67)(cid:66)(cid:77)(cid:70)(cid:84)(cid:1)(cid:80)(cid:71)(cid:1)(cid:73)(cid:66)(cid:90)(cid:1)
(cid:74)(cid:79)(cid:1)(cid:66)(cid:1)(cid:84)(cid:82)(cid:86)(cid:66)(cid:83)(cid:70)(cid:1)(cid:77)(cid:80)(cid:71)(cid:85)(cid:1)(cid:74)(cid:71)(cid:1)(cid:85)(cid:73)(cid:70)(cid:1)(cid:84)(cid:74)(cid:69)(cid:70)(cid:1)(cid:80)(cid:71)(cid:1)(cid:85)(cid:73)(cid:70)(cid:1)(cid:77)(cid:80)(cid:71)(cid:85)(cid:1)(cid:88)(cid:66)(cid:84)(cid:1)(cid:85)(cid:83)(cid:74)(cid:81)(cid:77)(cid:70)(cid:69)(cid:1)(cid:74)(cid:79)(cid:1)(cid:77)(cid:70)(cid:79)(cid:72)(cid:85)(cid:73)(cid:15)(cid:1)(cid:41)(cid:80)(cid:88)(cid:70)(cid:87)(cid:70)(cid:83)(cid:13)(cid:1)(cid:85)(cid:73)(cid:70)(cid:90)(cid:1)(cid:69)(cid:74)(cid:69)(cid:1)(cid:79)(cid:80)(cid:85)(cid:1)(cid:76)(cid:79)(cid:80)(cid:88)(cid:1)(cid:73)(cid:80)(cid:88)(cid:1)(cid:85)(cid:80)(cid:1)
(cid:68)(cid:66)(cid:77)(cid:68)(cid:86)(cid:77)(cid:66)(cid:85)(cid:70)(cid:1)(cid:85)(cid:73)(cid:70)(cid:1)(cid:77)(cid:70)(cid:79)(cid:72)(cid:85)(cid:73)(cid:1)(cid:80)(cid:71)(cid:1)(cid:85)(cid:73)(cid:70)(cid:1)(cid:84)(cid:74)(cid:69)(cid:70)(cid:1)(cid:80)(cid:71)(cid:1)(cid:66)(cid:1)(cid:84)(cid:82)(cid:86)(cid:66)(cid:83)(cid:70)(cid:1)(cid:84)(cid:85)(cid:66)(cid:83)(cid:85)(cid:74)(cid:79)(cid:72)(cid:1)(cid:71)(cid:83)(cid:80)(cid:78)(cid:1)(cid:66)(cid:1)(cid:72)(cid:74)(cid:87)(cid:70)(cid:79)(cid:1)(cid:66)(cid:83)(cid:70)(cid:66)(cid:15)(cid:1)(cid:53)(cid:73)(cid:70)(cid:1)(cid:88)(cid:80)(cid:83)(cid:69)(cid:1)(cid:105)(cid:82)(cid:86)(cid:66)(cid:69)(cid:83)(cid:66)(cid:85)(cid:74)(cid:68)(cid:119)(cid:1)
(cid:68)(cid:80)(cid:78)(cid:70)(cid:84)(cid:1)(cid:71)(cid:83)(cid:80)(cid:78)(cid:1)(cid:105)(cid:82)(cid:86)(cid:66)(cid:69)(cid:83)(cid:66)(cid:85)(cid:86)(cid:84)(cid:13)(cid:119)(cid:1)(cid:85)(cid:73)(cid:70)(cid:1)(cid:45)(cid:66)(cid:85)(cid:74)(cid:79)(cid:1)(cid:88)(cid:80)(cid:83)(cid:69)(cid:1)(cid:71)(cid:80)(cid:83)(cid:1)(cid:105)(cid:84)(cid:82)(cid:86)(cid:66)(cid:83)(cid:70)(cid:15)(cid:119)(cid:1)(cid:42)(cid:79)(cid:1)(cid:85)(cid:73)(cid:74)(cid:84)(cid:1)(cid:68)(cid:73)(cid:66)(cid:81)(cid:85)(cid:70)(cid:83)(cid:13)(cid:1)(cid:88)(cid:70)(cid:1)(cid:88)(cid:74)(cid:77)(cid:77)(cid:1)(cid:77)(cid:70)(cid:66)(cid:83)(cid:79)(cid:1)(cid:73)(cid:80)(cid:88)(cid:1)(cid:85)(cid:80)(cid:1)
(cid:84)(cid:80)(cid:77)(cid:87)(cid:70)(cid:1)(cid:68)(cid:70)(cid:83)(cid:85)(cid:66)(cid:74)(cid:79)(cid:1)(cid:82)(cid:86)(cid:66)(cid:69)(cid:83)(cid:66)(cid:85)(cid:74)(cid:68)(cid:1)(cid:70)(cid:82)(cid:86)(cid:66)(cid:85)(cid:74)(cid:80)(cid:79)(cid:84)(cid:1)(cid:67)(cid:90)(cid:1)(cid:71)(cid:66)(cid:68)(cid:85)(cid:80)(cid:83)(cid:74)(cid:79)(cid:72)(cid:1)(cid:81)(cid:80)(cid:77)(cid:90)(cid:79)(cid:80)(cid:78)(cid:74)(cid:66)(cid:77)(cid:84)(cid:15)
1
1 The
(cid:18)(cid:15) (cid:1) (cid:1)(cid:40)(cid:83)(cid:70)(cid:66)(cid:85)(cid:70)(cid:84)(cid:85)(cid:1)(cid:36)(cid:80)(cid:78)(cid:78)(cid:80)(cid:79)(cid:39)(cid:66)(cid:68)(cid:85)(cid:80)(cid:83)
We begin this section with definitions of factors and divisors. Because 24 =
2·12, both 2 and 12 are factors of 24. However, note that 2 is also a divisor
of 24, because when you divide 24 by 2 you get 12, with a remainder of zero.
Similarly, 12 is also a divisor of 24, because when you divide 24 by 12 you get
2, with a remainder of zero.
Factors and divisors. Suppose m and n are integers. Then m is a divisor
(factor) of n if and only if there exists another integer k so that n=m·k.
The words divisor and factor are equivalent. They have the same meaning.
You Try It!
List the positive divisors of EXAMPLE 1. List the positive divisors (factors) of 24.
18.
Solution: First, list all possible ways that we can express 24 as a product of
two positive integers:
24=1·24 or 24=2·12 or 24=3·8 or 24=4·6
Answer: 1,2,3,6,9,and 18 Therefore, the positive divisors (factors) of 24 are 1, 2, 3, 4, 6, 8, and 24.
(cid:2)
You Try It!
Listthe positive divisorsthat EXAMPLE 2. List the positive divisors (factors) that 36 and 48 have in
40 and 60 have in common. common.
Solution: First,listallpositivedivisors(factors)of36and48separately,then
box the divisors that are in common.
Divisors of 36 are: 1 , 2 , 3 , 4 , 6 ,9, 12 ,18,36
Divisors of 48 are: 1 , 2 , 3 , 4 , 6 ,8, 12 ,16,24,48
Therefore, the common positive divisors (factors) of 36 and 48 are 1, 2, 3, 4,
Answer: 1,2,4,5,10,and 20 6, and 12.
(cid:2)
2
(cid:18)(cid:15)(cid:18)(cid:1)(cid:53)(cid:73)(cid:70)(cid:1)(cid:40)(cid:83)(cid:70)(cid:66)(cid:85)(cid:70)(cid:84)(cid:85)(cid:1)(cid:36)(cid:80)(cid:78)(cid:78)(cid:80)(cid:79)(cid:1)(cid:39)(cid:66)(cid:68)(cid:85)(cid:80)(cid:83)
Greatest common divisor. Thegreatestcommondivisor(factor)ofaandb
isthelargestpositivenumberthatdividesevenly(noremainder)bothaandb.
ThegreatestcommondivisorofaandbisdenotedbythesymbolismGCD(a,b).
WewillalsousetheabbreviationGCF(a,b)torepresentsthegreatestcommon
factor of a and b.
Remember, greatest common divisor and greatest common factor have the
samemeaning. InExample 2,welistedthecommonpositivedivisorsof36and
48. The largestofthese commondivisorswas12. Hence, the greatestcommon
divisor (factor) of 36 and 48 is 12, written GCD(36,48)=12.
With smaller numbers, it is usually easy to identify the greatest common
divisor (factor).
You Try It!
EXAMPLE 3. State the greatest common divisor (factor) of each of the State the greatest common
following pairs of numbers: (a) 18 and 24, (b) 30 and 40, and (c) 16 and 24. divisor of 36 and 60.
Solution: Ineachcase, we must find the largestpossible positive integer that
divides evenly into both the given numbers.
a) The largest positive integer that divides evenly into both 18 and 24 is 6.
Thus, GCD(18,24)=6.
b) The largest positive integer that divides evenly into both 30 and 40 is 10.
Thus, GCD(30,40)=10.
c) The largest positive integer that divides evenly into both 16 and 24 is 8.
Thus, GCD(16,24)=8.
Answer: 12
(cid:2)
With larger numbers, it is harder to identify the greatest common divisor
(factor). However, prime factorization will save the day!
You Try It!
EXAMPLE 4. Find the greatest common divisor (factor) of 360 and 756. Find the greatest common
divisor of 120 and 450.
Solution: Primefactor360and756,writingyouranswerinexponentialform.
3
(cid:36)(cid:73)(cid:66)(cid:81)(cid:85)(cid:70)(cid:83)(cid:1)(cid:18)(cid:1)(cid:39)(cid:66)(cid:68)(cid:85)(cid:80)(cid:83)(cid:74)(cid:79)(cid:72)
360 756
36 10 9 84
4 9 2 5 3 3 7 12
2 2 3 3 3 4
2 2
Thus:
360=23·32·5
756=22·33·7
To find the greatest common divisor (factor), list each factor that appears in
common to the highest power that appears in common.
In this case, the factors 2 and 3 appear in common, with 22 being the high-
est power of 2 and 32 being the highest power of 3 that appear in common.
Therefore, the greatest common divisor of 360 and 756 is:
GCD(360,756)=22·32
=4·9
=36
Therefore, the greatest common divisor (factor) is GCD(360,756)= 36. Note
what happens when we write each of the given numbers as a product of the
greatest common factor and a second factor:
360=36·10
756=36·21
In each case, note how the second second factors (10 and 21) contain no addi-
Answer: 30 tional common factors.
(cid:2)
Finding the Greatest Common Factor of Monomials
Example 4 reveals the technique used to find the greatest common factor of
two or more monomials.
Finding the GCF of two or more monomials. To find the greatest com-
mon factor of two or more monomials, proceed as follows:
4
(cid:18)(cid:15)(cid:18)(cid:1)(cid:53)(cid:73)(cid:70)(cid:1)(cid:40)(cid:83)(cid:70)(cid:66)(cid:85)(cid:70)(cid:84)(cid:85)(cid:1)(cid:36)(cid:80)(cid:78)(cid:78)(cid:80)(cid:79)(cid:1)(cid:39)(cid:66)(cid:68)(cid:85)(cid:80)(cid:83)
1. Find the greatestcommonfactor (divisor)of the coefficients of the given
monomials. Use prime factorization if necessary.
2. List each variable that appears in common in the given monomials.
3. Raise each variable that appears in common to the highest power that
appears in common among the given monomials.
You Try It!
EXAMPLE 5. Find the greatest common factor of 6x3y3 and 9x2y5. Find the greatest common
factor of 16xy3 and 12x4y2.
Solution: To find the GCF of 6x3y3 and 9x2y5, we note that:
1. The greatest common factor (divisor) of 6 and 9 is 3.
2. The monomials 6x3y3 and 9x2y5 have the variables x and y in common.
3. The highest power of x in common is x2. The highest power of y in
common is y3.
Thus, the greatest common factor is GCF(6x3y3,9x2y5) = 3x2y3. Note what
happenswhenwewriteeachofthegivenmonomialsasaproductofthegreatest
common factor and a second monomial:
6x3y3 =3x2y3·2x
9x2y5 =3x2y3·3y
Observe that the set of second monomial factors (2x and 3y) contain no addi-
tional common factors. Answer: 4xy2
(cid:2)
You Try It!
EXAMPLE 6. Find the greatest common factor of 12x4, 18x3, and 30x2. Find the greatest common
factor of 6y3, 15y2, and 9y5.
Solution: To find the GCF of 12x4, 18x3, and 30x2, we note that:
1. The greatest common factor (divisor) of 12, 18, and 30 is 6.
2. The monomials 12x4, 18x3, and 30x2 have the variable x in common.
3. The highest power of x in common is x2.
5
(cid:36)(cid:73)(cid:66)(cid:81)(cid:85)(cid:70)(cid:83)(cid:1)(cid:18)(cid:1)(cid:39)(cid:66)(cid:68)(cid:85)(cid:80)(cid:83)(cid:74)(cid:79)(cid:72)
Thus, the greatestcommonfactor is GCF(12x4,18x3,30x2)=6x2. Note what
happenswhenwewriteeachofthegivenmonomialsasaproductofthegreatest
common factor and a second monomial:
12x4 =6x2·2x2
18x3 =6x2·3x
30x2 =6x2·5
Observe that the set of second monomial factors (2x2, 3x, and 5) contain no
Answer: 3y2 additional common factors.
(cid:2)
Factor Out the GCF
In Chapter 5, we multiplied a monomial and polynomial by distributing the
monomial times each term in the polynomial.
2x(3x2+4x−7)=2x·3x2+2x·4x−2x·7
=6x3+8x2−14x
Inthissectionwereversethatmultiplicationprocess. We presentyouwiththe
final product and ask you to bring back the original multiplication problem.
In the case 6x3+8x2−14x, the greatest common factor of 6x3, 8x2, and 14x
is 2x. We then use the distributive property to factor out 2x from each term
of the polynomial.
6x3+8x2−14x=2x·3x2+2x·4x−2x·7
=2x(3x2+4x−7)
Factoring. Factoring is “unmultiplying.” You are given the product, then
asked to find the original multiplication problem.
First rule of factoring. If the terms of the givenpolynomial have a greatest
common factor (GCF), then factor out the GCF.
Let’s look at a few examples that factor out the GCF.
You Try It!
Factor: 9y2−15y+12 EXAMPLE 7. Factor: 6x2+10x+14
6
(cid:18)(cid:15)(cid:18)(cid:1)(cid:53)(cid:73)(cid:70)(cid:1)(cid:40)(cid:83)(cid:70)(cid:66)(cid:85)(cid:70)(cid:84)(cid:85)(cid:1)(cid:36)(cid:80)(cid:78)(cid:78)(cid:80)(cid:79)(cid:1)(cid:39)(cid:66)(cid:68)(cid:85)(cid:80)(cid:83)
Solution: The greatestcommonfactor (GCF) of6x2, 10xand14is 2. Factor
out the GCF.
6x2+10x+14=2·3x2+2·5x+2·7
=2(3x2+5x+7)
Checking your work. Every time you factor a polynomial, remultiply to
check your work.
Check: Multiply. Distribute the 2.
2(3x2+5x+7)=2·3x2+2·5x+2·7
=6x2+10x+14
That’s the original polynomial, so we factored correctly. Answer: 3(3y2−5y+4)
(cid:2)
You Try It!
EXAMPLE 8. Factor: 12y5−32y4+8y2 Factor: 8x6+20x4−24x3
Solution: The greatest common factor (GCF) of 12y5, 32y4 and 8y2 is 4y2.
Factor out the GCF.
12y5−32y4+8y2 =4y2·3y3−4y2·8y2+4y2·2
=4y2(3y3−8y2+2)
Check: Multiply. Distribute the monomial 4y2.
4y2(3y3−8y2+2)=4y2·3y3−4y2·8y2+4y2·2
=12y5−32y4+8y2
That’s the original polynomial. We have factored correctly. Answer: 4x3(2x3+5x−6)
(cid:2)
You Try It!
EXAMPLE 9. Factor: 12a3b+24a2b2+12ab3 Factor:
15s2t4+6s3t2+9s2t2
Solution: The greatest common factor (GCF) of 12a3b, 24a2b2 and 12ab3 is
12ab. Factor out the GCF.
12a3b+24a2b2+12ab3=12ab·a2−12ab·2ab+12ab·b2
=12ab(a2+2ab+b2)
7