Polynomial Equations
A polynomial equation in two variables is an equation of the form
p(x, y) = q(x, y)
where both p(x, y) and q(x, y) are polynomials in two variables.
Examples.
• xy + 2 = y 2
3x
4
(xy + 2 is a quadratic polynomial. So is y 2
•y
3x
4.)
x=2
(y
• x2
• 3x2
x is a linear polynomial. 2 is a constant polynomial.)
5x + y
2=
xy + 4y 2
7xy
y+2
5x + 6
7=0
Domains of polynomial equations
Because every polynomial in two variables has a domain of R2 , the implied
domain of any polynomial equation in two variables is R2 , the entire plane.
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Solutions of equations
If p(x, y) = q(x, y) is a polynomial equation in two variables, then a point
in the plane (↵, ) 2 R2 is a solution of the equation if the number p(↵, )
equals the number q(↵, ). That is, if p(↵, ) = q(↵, ).
Examples.
• The point in the plane (1, 2) is a solution of the equation
x2 + y 2 = x + y + 2 because 12 + 22 = 5 = 1 + 2 + 2.
• The point (0, 1) is a solution of the same equation x2 +y 2 = x+y +2
because 02 + ( 1)2 = 1 = 0 + ( 1) + 2.
119
• The point (3, 1) is not a solution of the same equation
x•2 +
y2 =
x +(3,
y +1)2 isbecause
32 + 12 = 10 6= 6 = 3 + 1 + 2.
The
point
not a solution
of the same equation
32+12 = 106=3+i+2.
because
+
x
2
y =x+y+2
The set of
every
solution of an
equation
p(x, y) = q(x, y) is the set
The set of
S = {solutions
(↵, ) 2 Rof2 an
| p(↵,
) = q(↵,
equation
p(x, y)) }=
q(x, y)
is the set
Thus, if S is the set of
solutions ofER
the equation
from the previous examples,
S={()
2
p(3)=q(3)}
2
2
x + Thus,
y = xif+Syis+the
2, set
thenof (1,
2) 2 S and (0, 1) 2 S, while (3, 1) 2
/ S.
solutions of the equation from the previous examples,
= x+y+2, then (1,2) E Sand (0,—i) eS, while (3,1)ØS.
Solutions
as geometric objects
The
set of solutions
of a polynomial
equation in one variable is always finite,
Solutions
as geometric
objects
so we can just
write out a list of the solutions. In contrast, a polynomial
of solutions of a polynomial eauatirrtiiio variables is a subset of
equation
in two variables can have infinitely many solutions. It would be
the plane.
:aw the set of solutions of a particular
impossible
to
write
a
list
of
infinitely
many
points in the plane, but it often
equation.
1is o
or.
is possible to make a drawing of every point in the planea that is a solution of
s.4 ti
a particular
equation.
Examples.
Examples.
• Let p(x, y) be the linear polynomial x 3 and let q(x, y)e
constant polynomial 0. The solutions of the equation p(r, y) = q(x, y), the+Zc cç
• Let p(x, y) be the linear polynomial x and let q(x, y) be the constant
4
equation x 3 = 0, are the points (, 3) in the plane with the property thatC
polynomial 3. The solutions of the equation p(x, y) = q(x, y), the equationv..tiy (Whcq
Lt
x = 3, are the points (↵, ) in the plane with the property that
wSS
or more simply, points↵ (a,
in
the
plane
with
/) ) = q(↵, ) = 3 o = 3. Thus, (3, 7), and
= p(↵,
(3, —2), and (3, 100) are solutions of the equation x
0. The set of
Thus, (3, 7), and (3, 2), and (3, 100) are solutions of the equation x = 3.
all solutions, of all pairs of numbers whose x-coordina’qual 3, forms a
The set of all solutions, of all pairs of numbers whose x-coordinates equal 3,
vertical line in the plane.
forms a vertical line in the plane.
—
—
—
S
SoLA+;or1s
o x-O
120
116
• Similar
to to
thethe
previous
thethe
setset
of of
solutio
• Similar to the previous example,
the set
of
solutions
ofexample,
the
equation
• Similar
previous
example,
solut
forms
5=
0 also
vertiline
in in
thethe
plane.
x = 5 also forms a vertical linexinx the
It’s
a avertical
line comprised
of It’sIt’s
a verti
forms
5 plane.
=
0 also
a vertiline
plane.
a ve
points
of of
whose
x-coordinat’equal
5. 5.
points whose x-coordinates equal
5.points
whose
x-coordinat’equal
—
—
55
• x• =
anan
equation
inofin
two
• x = y is an equation in two variables.
solutions
this
equation
variables.
solution
y isy The
x =
is
equation
two
variables.The
The
solut
areare
allallof
the
/3)words,
(ce,(ce,
are all of the points (↵, ) such
that
↵ of
= thepoints
. points
In other
thethat
solutions
o o= =/3. /3.In Inother
/3)such
suchthat
otherwo
this
areare
allall
of of
thethe
of this equation are all of the of
points
of equation
the form
(↵,
↵)
—
all
of the
points
of points
thetheform
c) c)
of
thisequation
points
of
form(ce,(ce,
whose
x-coordinates
equal
whose x-coordinates equal their
y-coordinates.
There
aretheir
infinitely
many of There
y-coordinates.
whose
x-coordinates
equal
their
y-coordinates.
Therearear
• Similar to the previous example, the set of solutions of the equation
these
solutions,
including
7) 7)and
(—10,
—10).
these solutions, including (4, 4),
(7, 7)
and ( 10,
10). (4,4),
If (4,4),
we placed
a and
tiny
(7,(7,
these
solutions,
including
(—10,
—10).If
x 5 = 0 also forms a vertiline in the plane. It’s a vertical line comprised
in in
thethe
plane
dot in the plane for each of these
solutions,
they’d
collectively
form
a line,they’d
a they’d
dotdot
forfor
each
of of
these
solutions,
collecti
plane
each
these
solutions,
collec
of points whose x-coordinat’equal 5.
line
that’s
line that’s often called the “x =
yline
line”.
often
called
thethe“x “x
y line”.
that’s
often
called
y line”.
= =
—
5
x = y is an equation in two2 variables.2 The solutions of this equation
• The set of •solutions
of the equation
4x
2xy
+ y of
+ of
3x
7yequation
5 =4x
0 4x
• The
of of
set
solutions
thethe
equation
2
• The
2
y
set
solutions
++
2 2xy
2xy
are all of the points (ce, /3) such that o = /3. In other words, the solutions
form a geometric object called
angeometric
ellipse.
We’ll
have
more
to
sayWe’ll
about
isa
object
called
ananellipse.
have
more
to
isa
geometric
object
called
ellipse.
We’ll
have
more
of this equation are all of the points of the form (ce, c)
of
the
points
all
ellipses later. They’re exampleslater.
of
conics.
They’re
examples
of of
conics.
later.
They’re
examples
conics.
whose x-coordinates equal their y-coordinates. There are infinitely many of
these solutions, including (4,4), (7, 7) and (—10, —10). If we placed a tiny
dot in the plane for each of these solutions, they’d collectively form a line, a
line that’s often called the “x = y line”.
—
—
qx
2
2
÷+3x-75
2
2
qx
÷+3x-75o
121
117 117
4
• •The
Theset
solutionsofofthe
setofofsolutions
equation yy
the equation
4
devil’s curve”.
4“devil’s
curve’
—
=
y2 =
x4
—
2x2
2x2 isis callee
called the
‘I
N
i.
• •The
Thethree
threeprevious
previousexamples
exampleswere
wereofofequations
thathad
hadinfinitely
equationsthat
infinitely
many
manysolutions.
solutions.Sometimes
Sometimesa apolynomial
polynomialequation
equationwill
willhave
haveno
nosolutions.
solutions. IfIf
you
yousquare
squarea anumber,
number,the
theresult
resultcannot
cannotbebenegative.
negative.IfIfyou
youadd
addtwo
twononnegnonneg
ative
ativenumbers
numberstogether,
together,the
theresult
resultisisstill
stillnonnegative.
nonnegative.This
Thisisistotosay
saythat
thatthe
the
2
2
equation
1 has
equationx x
2++y y
hasnonosolution.
solution.There
Thereare
arenonopairs
pairsofofnumbers
2= —1
numbersthat
that
you
can
square
and
then
add
together
to
get
the
negative
number
1.
you can square and then add together to get the negative number —1.
• •Some
Someequations
equationsinintwo
twovariables
variableshave
havea afinite
finitenumber
numberofofsolutions.
solutions.
The
claim
below
displays
one
such
equation.
The claim below displays one such equation.
Claim:
0).
Claim:The
Theonly
onlysolution
solutionofofthe
theequation
equationx2x
2++y 2y
thepoint
point(0,
2==0 0isisthe
(0,0).
—.
‘(0,0)
2
Proof:
++y 2y
Proof:Suppose
Supposethat
that(↵,
,B)isisa asolution
solutionofofthe
theequation
equationxx
2
We’ll
2==0.0. We’ll
(a’, )
show
0). That
0) isisthe
showthat
that(↵,
Thatmeans
3)==(0,(0,0).
that(0,
meansthat
theonly
onlysolution.
solution.
(0,0)
(a’, )
22
2
2
2
Ifi(↵,
solutionofofthe
theequation
equationx ++y y
then↵ ++ 2 ==0.0. The
The
2==0,0,then
ta’, )i’isisa asolution
2
2
/32
square
0.0.Since
squareofofa anumber
numbercan’t
can’tbehenegative.
negative.Thus
Thus↵a’
and
2 0 0and
Sinceneither
neither
2
2,2
↵ a’
are
negative,
the
only
way
they
could
sum
to
0
is
if
they
both
2nor
nor
are negative, the only way they could sum to 0 is if they bothequal
equal
2
2/32
2
2
32
zero.
++ ==0 0implies
zero.That
Thatis,is,↵a’
2
impliesthat
that↵a’
2==0 0and
and ==0.0.The
Theonly
onlynumber
number
122
118
that you can square to get 0, is 0 itself. In other words, since ↵2 = 0, we
must have that ↵ = 0. Since 2 = 0, we must have that = 0.
We’ve now shown what we wanted to. If (↵, ) is a solution of the equation
2
x + y 2 = 0, then ↵2 + 2 = 0, which implies that (↵, ) = (0, 0). Thus, (0, 0)
is the only solution of the equation x2 + y 2 = 0.
⌅
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Equivalent equations
Similar to equations in one variable, there are rules for when equations in
two variables are equivalent, and equivalent equations have the same solutions. When it comes to polynomial equations in two variables, there are
only two rules for equivalent equations that we’ll need.
Equivalent by addition:
The equation p(x, y) + h(x, y) = q(x, y)
is equivalent to p(x, y) = q(x, y) h(x, y).
Equivalent by multiplication:
If c 6= 0 is a number, then the equation cp(x, y) = q(x, y)
is equivalent to p(x, y) = q(x,y)
c .
Consequence of equivalent equations:
Equivalent equations have the same set of solutions.
Example.
• The equation
8x2 + 2y 2 + 6x = 4xy + 14y + 10
123
Similar to the previous example, the set of solutions of the equation
5 = 0 also forms a verticQine in the 5plane. It’s a vertical line comprised
points whose x-coordiuaiequal 5.
x5O
is equivalent by addition to the equation
. x =
2 y is an equation in two variables. The solutions of this equa
8x2 + 2y
+ 6x (4xy + 14y + 10) = 0
are all of the points (ce, ,8) such that c = /3. In other words, the solut
thisterms
We
aboveare
to all
write
this points
equation
as form (cu, c)
of the
equation
of the
all of the po
5 can rearrangeofthe
whose2 x—coordinates
2 equal their y-coordinates. T•here are infinitely man
8x solutions,
4xy + 2y
+ 6x (4,4),
14y 10
x5O these
including
(7, =
7) 0and (—10, —10). If we placed a
the plane2 for
solutions, as
of these
they’d collectively form a li
We can factor outdot
theinconstant
andeach
rewrite
this equation
line that’s often called the “ = y line”.
. x = y is an equation in two2 variables.2 The solutions
2(4x
2xy + y + 3x 7y 5) =of0 this equation
all of the points (ce, ,8) such that c = /3. In other words, the solutions
Thisare
is equivalent
multiplication
to the(cu,
equation
his equation
all of theby
points
of the form
all of the points
c)
ose x—coordinates equal their y-coordinates. T•here are infinitely
many of
0
2
2
4x
2xy
+
y
+
3x
7y
5
=
=
0
se solutions, including (4,4), (7, 7) and (—10, —10). If we placed a tiny
2
in the plane for each of these solutions, they’d collectively form a line, a
We called
saw thethe
solutions
of this equation earlier in the chapter. The solutions
that’s often
“ = y line”.
form an ellipse. Because equivalent equations have the same set of solutions,
the ellipse is also the set of solutions of our original equation from this example
—
—
2 of solutions of the equation 4x
2 2xy + y
5
2 + 3x
8x. 2The
+ 2yset
+ 6x = 4xy + 14y + 10
geometric object called an ellipse. We’ll have more to say about ell
later. They’re examples of conics.
—
—
—
Zi
z
. The set of solutions of the equation 4x
2
2xy + y
5 = 0
2 + 3x
geometric object called an ellipse. We’ll have
more to say about ellipses
+1L+1O
er. They’re examples of conics.
—
—
—
117
Zi
z
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*
+1L+1O
Planar transformations
of solutions
117
Solutions of polynomial equations are geometric objects, and at times, we’ll
be interested in how planar transformations a↵ect these geometric objects.
The general principal is described in the following claim.
124
Claim: Suppose T : R2 ! R2 is a planar transformation — either an
addition function or an invertible matrix — and that S ✓ R2 is the set of
solutions of the polynomial equation p(x, y) = q(x, y). Then T (S) is the set
of solutions of the equation p T 1 (x, y) = q T 1 (x, y).
Proof: Let’s take a point in the set T (S). That is, a point of the form
T (↵, ) where (↵, ) 2 S, which means that (↵, ) is a solution of the equation
p(x, y) = q(x, y), or in other words, that
p(↵, ) = q(↵, )
We want to show that this point, T (↵, ), in the set T (S) is a solution of the
equation
p T 1 (x, y) = q T 1 (x, y)
To see this, we need to check that p T 1 (T (↵, )) equals q T 1 (T (↵, )).
We’ll do this in a moment, but before we do, remember two things: that
the definition of inverse functions is that T 1 (T (x, y)) = (x, y), and that
p(↵, ) = q(↵, ). Now we check that p T 1 (T (↵, )) equals q T 1 (T (↵, )):
1
(T (↵, )) = p(T
S
p T
1
(T (↵, )))
= p(↵, )
oç
1
cuo
= q(↵, )
= q(T
=q T
1
(T (↵, )))
1
(T (↵, ))
Thus, points in the set T (S), such as T (↵, ), are solutions of the equation
p T 1 (x, y) = q T 1 (x, y).
⌅
The previous claim is important to stress. We’ll call it the Principle of
Transforming Solutions (POTS). If S is the set of solutions of p(x, y) =
q(x, y), then T (S) is the set of solutions of p T 1 (x, y) = q T 1 (x, y).
S
cLo(R’Loa
125
:S’AOQS
Example.
• Let p(x, y) = x and q(x, y) = 3. If S is the set of solutions of the
equation p(x, y) = q(x, y), which is the equation x = 3, then we’ve seen that
S is a vertical line.
,o) : R
2
The addition function A(
2 —+ R moves ponts in the plane right by
The addition function A(2,0) : R2 ! R2 moves points in the plane right by
.o)(S) is the line of so1utions — 3 = 0 shifted right by
2
2 units, so the set A(
2 units, so the set A(2,0) (S) is the line of solutions of x = 3 shifted right by 2
2 units. It’s the vertical line of points whose x-coordinatqua1.’5.
units. It’s the vertical line of points whose x-coordinates equal 5.
-
S
3
S
POTS tells us that the line A(2,0) (S) is the set of solutions of the equation
1
1
Namely,
p A(2,0)
(x, y) = q A(2,0)
(x, y)
,o) (S), is the set of solutions
2
A(
of the equation
1
Because A(2,0)
(x, y) = (x 2, y), you can check that the equation
poA’O)(x,y)
=0
1
1
p A(2,0)
(x, y) = q A(2,0)
(x, y)
In order to progress further, we’d need to know what the inverse of the
is the same asA(
the equation
planar transformation
,o)
2
is, and we do: A’
) = A_(
0
,o). Thus, our new
2
x
2
=
3
line, A(
,o) (8), is the set of solutions of the equation
2
In summary, POTS tells us that the line A(2,0) (S) is the set of solutions of
,o)(x, y) = 0
2
p o A_(
x 2 = 3, or equivalently,
of x = 5.
We want to write the above equation more simply. To this end, notice that
o
,o)(x, y)
2
A(_
=
p(x
—
2, y)
(x
2)
3
x
5
r
p
=
—
—
=
—
I
Thus, theVA(
,o)(S)is the set of solutions of the equation x — 5 = 0. We
2
had seen lier in this chapter that this is indeed the case. The solutions of
x — 5 = 0 are a vertical line of points whose x-coordinate equals 5.
In-i
126
If S is the set of solutions of p(x, y) = 0,
then T(S) is the set of solutions of p o T
(x, y)
1
122
=
0.
U
J%J
The statement of the previous claim is important -po4ii- to stress. We’ll
use it to help us determine what sorts of solutions there are to equations
p(x, y) = 0 when p(x, y) is a quadratic polynomial.
S
:S’AOQS
POTS
This is very important. It’s worth repeating.
cLo(R’Loa
If S is the set of solutions of p(x, y) = q(x, y),
then T (S) is the set of solutions of p T 1 (x, y) = q T
1
(x, y).
S
(5)1
ç
0
(R’
127
Exercises
Let’s look at the equation 5x + y 2 = 3x2 xy. Determine whether the
points given in #1-4 are solutions of this equation.
1.) (2, 3)
2.) (1, 0)
3.) ( 1, 2)
5.) Are the equations x2 + 2 = x and x2
6.) Are the equations xy = y 2 and xy
7.) Are xy + 3 = y 2
x2 and 3 = y 2
8.) Are the equations 2x
4.) (3, 4)
x + 2 = 0 equivalent by addition?
x = y2
x2
y equivalent by addition?
xy equivalent by addition?
3 = x2 and 2x = x2 equivalent by addition?
9.) Are the equations 2xy 3x2 = y 2
equivalent by multiplication?
x
10.) Are 5x2
plication?
x + 2 = 4y 2 equivalent by multi-
5x + 10 = 20y 2 and x2
y and xy
3x2 = y 2
x
y
We saw in this chapter that x2 +y 2 = 0 has a single solution, the point (0, 0).
In #11-14, let p(x, y) = x2 + y 2 and q(x, y) = 0, so that p(x, y) = q(x, y) is
an equation whose only solution is (0, 0).
11.) What is A(2,3) (0, 0)?
1
1
12.) What are p A(2,3)
(x, y) and q A(2,3)
(x, y)?
13.) What’s the only solution of the polynomial equation
1
1
p A(2,3)
(x, y) = q A(2,3)
(x, y)? (You can use POTS and #11 to answer this.)
14.) Use POTS and the polynomials p(x, y) = x2 + y 2 and q(x, y) = 0 (as
in #13) to write a polynomial equation in two variables that has only one
solution: the point (a, b) in the plane.
128
loge (x) is the most common logarithm used in math. There are lots of
benefits to using logarithms base e, and these benefits will be explained in
calculus. Because of these benefits, some call logarithm base e the “natural
logarithm”, and they write it as ln(x). (Scientists often prefer to write ln(x).
Mathematicians often write loge (x) as log(x). To make matters more confusing, if a calculator has a button for log(x), it probably means log10 (x).)
Because plenty of people write loge (x) as ln(x), we should practice seeing and
writing the logarithm base e in this way. Find the values asked for in #15-30.
15.) ln 1 (2)
16.) ln(e)
17.) ln(e2 )
18.) ln(e3 )
19.) ln(e4 )
20.) ln( 1e )
21.) ln( e12 )
22.) ln( e13 )
p
23.) ln( e)
p
24.) ln( 3 e)
p
25.) ln( 4 e)
26.) ln( p1e )
27.) ln( p31e )
28.) ln( p41e )
29.) ln(1)
30.) ln 1 (e)
Find the solutions of the following equations in one variable.
31.) ln(2x
3) = 0
32.) ln(3x
5) = 2
129
33.) 3 ln(7
x)
4=
5