Introduction
Introduction
Thistext
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ofsome
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This
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will
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with
brief
review
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precalculus
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that
This
youare
areassumed
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comfortablewith:
with:quadratic
quadraticpolynomials,
polynomials,exponenexponenyou
are
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comfortable
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quadratic
polynomials,
exponenyou
tials,and
andlogarithms,
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asa aafew
fewexamples.
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tials,
and
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examples.
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then
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brief
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tials,
tolinear
linearalgebra
algebrainin
inthe
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Thisisisisthe
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studyofof
ofvectors
vectorsinin
inthe
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ofthe
the
linear
algebra
the
plane.
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the
study
vectors
the
plane,
the
toto
algebrarelating
relatingtoto
tovectors,
vectors,and
andofof
ofmatrices,
matrices,which
whichare
arethe
themost
mostimportant
important
algebra
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and
matrices,
which
are
the
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functionsthat
thatare
areused
usedtoto
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toeach
eachother.
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Linearalgebra
algebraisis
isnot
notthe
theprimary
primaryinterest
interestofof
ofthis
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somebasic
basiclinear
linearalgebra
algebrainin
inorder
ordertoto
tobetter
betterstudy
studythe
thetopics
topicsthat
thatthis
thiscourse
course
some
basic
linear
algebra
order
better
study
the
topics
that
this
course
some
isdevoted
devotedto,
to,namely,
namely,trigonometry,
trigonometry,conics,
conics,and
andthe
thecomplex
complexnumbers.
numbers.
devoted
to,
namely,
trigonometry,
conics,
and
the
complex
numbers.
isis
Trigonometryisis
isthe
thestudy
studyofof
oftriangles.
triangles.More
Moreprecisely,
precisely,itititisisisthe
thestudy
studyofof
of
Trigonometry
the
study
triangles.
More
precisely,
the
study
Trigonometry
therelationship
relationshipbetween
betweenthe
thelengths
lengthsofof
ofthe
thesides
sidesofof
ofa aatriangle
triangleand
andthe
theinterior
interior
the
relationship
between
the
lengths
the
sides
triangle
and
the
interior
the
anglesofof
ofa aatriangle.
triangle.
angles
triangle.
angles
bb
CC
2 22
As
As you’ve
you’ve probably
probably seen
seen before,
before, and
and as
as is
is drawn
drawn in
in the
the picture
picture above,
above, angles
angles
are
usually
drawn
as
circular
arcs.
This
is
because
angles
are
naturally
related
are usually drawn as circular arcs. This is because angles are naturally related
to
to the
the geometry
geometry of
of the
the circle,
circle, and
and while
while the
the historical
historical roots
roots of
of trigonometry
trigonometry
lie
in
the
study
of
triangles,
trigonometry
now
has
as
much
to
lie in the study of triangles, trigonometry now has as much to do
do with
with circircular
objects
as
it
does
with
triangles.
As
an
illustration
of
this,
we’ll
cular objects as it does with triangles. As an illustration of this, we’ll see
see
that
the
geometry
of
circular
motions,
called
rotations,
are
described
using
that the geometry of circular motions, called rotations, are described using
trigonometry.
trigonometry.
Conics
are
to
Conics
are solutions
solutions
to quadratic
quadratic equations
equations in
in two
two variables,
variables, such
such as
as
2
2
5x
2xy
+
y
6x
+
2y
+
3
=
0.
Conics
are
common
objects
in
2
2
5x
2xy + y
6x + 2y + 3 = 0. Conics are common objects in math
math
and
just
about
any
field
that
uses
math,
so
it’s
important
to
have
a
and just about any field that uses math, so it’s important to have a firm
firm
understanding
of
them.
We’ll
see
that
there
are
8
basic
types
of
conics:
those
understanding of them. We’ll see that there are 8 basic types of conics: those
that
that consist
consist of
of aa single
single point,
point, those
those that
that consist
consist of
of no
no points,
points, those
those that
that are
are
aa single
straight
line,
those
that
are
two
straight
lines,
and
those
that
are
single straight line, those that are two straight lines, and those that are
circles,
circles, ellipses,
ellipses, parabolas,
parabolas, or
or hyperbolas.
hyperbolas.
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Trigonometry
Trigonometry and
and the
the linear
linear algebra
algebra relating
relating to
to rotations
rotations will
will play
play an
an imimportant
role
in
leading
us
to
see
that
any
quadratic
equation
in
two
variables
portant role in leading us to see that any quadratic equation in two variables
is
is one
one of
of the
the 88 types
types above.
above.
3
3
p
4
-o
3
J
Complex numbers
numbers were
were either
either invented
invented or
or discovered,
discovered, depending
depending on
on your
your
Complex
preference of
of language,
language, in
in the
the 16th
16th century
century as
as aa means
means for
for finding
finding real
real number
number
preference
roots to
to cubic
cubic polynomials
polynomials with
with real
real number
number coefficients.
coefficients. More
More precisely,
precisely,
roots
what was
was discovered
discovered in
in the
the 16th
16th century
century was
was “the
“the cubic
cubic formula”,
formula”, which
which isis
what
counterpart to
to the
the quadratic
quadratic formula
formula that
that you
you already
already know.
know. The
The cubic
cubic
aa counterpart
formula requires
requires the
the use
use of
of complex
complex numbers
numbers to
to work
work it,
it, even
even ifif the
the formula
formula
formula
only used
used to
to find
find roots
roots that
that are
are real
real numbers.
numbers.
isis only
For example,
example, the
the cubic
cubic polynomial
polynomial xx33 15x
15x 44 has
has real
real number
number coefficoeffiFor
cients: 1,
1, 15,
15, and
and 4.
4. Using
Using the
the cubic
cubic formula,
formula, and
and the
the algebra
algebra of
of complex
complex
cients:
3
3
numbers, we
we can
can find
find
that xx p
15x 44 has
has three
three roots
roots which
which are
are all
all real
real
numbers,
15x
p that
p
p
1
1
numbers: 4,
4, 22+
+ 3,
3, and
and 22
3.
numbers:
3.
Later in
in this
this course
course itit will
will be
be explained
explained what
what the
the complex
complex numbers
numbers are
are
Later
exactly, but
but for
for now,
now, the
the complex
complex numbers
numbers are
are aa set
set of
of numbers
numbers that
that contain
contain
exactly,
the set
set of
of real
real numbers,
numbers, and
and that
that contain
contain many
many other
other numbers
numbers as
as well,
well, ininthe
cluding the
the most
most famous
famous example
example of
of aa complex
complex number,
number, the
the number
number i.i. The
The
cluding
2
defining equation
equation of
of the
the number
number ii isis that
that ii2 =
= 1.
1. Since
Since the
the square
square of
of any
any
defining
real number
number isis positive,
positive, we
we see
see that
that ii isis not
not aa real
real number.
number.
real
The connection
connection between
between the
the complex
complex numbers
numbers and
and trigonometry
trigonometry isis that,
that,
The
while addition
addition of
of complex
complex numbers
numbers isis best
best understood
understood using
using the
the geometry
geometry
while
of vectors,
vectors, multiplication
multiplication of
of complex
complex numbers
numbers isis best
best understood
understood using
using the
the
of
geometry of
of rotations.
rotations.
geometry
11There’s
There’s aa nice
nice account
account ofof this
this piece
piece ofof history
history in
in the
the chapter
chapter “Cardano
“Cardano and
and the
the Solution
Solution ofof the
the Cubic”
Cubic”
fromthe
thebook
book“Journey
“JourneyThrough
ThroughGenius”
Genius”by
byWilliam
WilliamDunham.
Dunham.
from
44
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Preparation for calculus
This is a precalculus text, and it is meant to prepare you for calculus. In
addition to learning the topics written about on the previous pages, that
preparation will come from learning the algebra that’s needed to succeed in a
calculus course. Occasionally we’ll take a break from the main themes of this
course to practice solving equations, using exponentials and logarithms, and
graphing functions, among other things. The most obvious examples of these
digressions that can be seen from the table of contents are the chapters titled
Equations in One Variable I-VII. These chapters are scattered throughout
the text, and a mastery of their contents will make learning calculus much
easier.
5
Exercises
The discriminant of a quadratic polynomial p(x) = ax2 + bx + c is b2
What are the discriminants of the following quadratic polynomials?
1.) 3x2
4ac.
4x + 7
2.) x2 + 5x
3.)
x2
4.)
4x2 + 2x
10
5
A number ↵ is a root of the polynomial p(x) if p(↵) = 0. If the discriminant
of a quadratic polynomial (with real number coefficients) is positive, then the
polynomial has two real numbers as roots. If it’s 0, then there’s only one root,
and if it’s negative, then there aren’t any real numbers that are roots. How
many real number roots do the following polynomials have?
5.)
2x2
6.)
x2 + 2x
7.) 5x2
3
1
x+1
8.) 2x2 + 3x
If the quadratic polynomial p(x) = ax2 +bx+c has real number coefficients,
and if the discriminant equals 0, then its only root is the number b/2a. If
the discriminant is positive, then there are two roots,
p
p
b + b2 4ac
b
b2 4ac
and
2a
2a
which is often abbreviated as
p
b ± b2 4ac
2a
What are the roots of the following polynomials?
9.) x2 + 4x + 1
10.) x2
11.)
x
12
2x2 + 4x
12.) 3x2
2x
2
2
6