Graphing Polynomials
In the previous chapter, we learned how to factor a polynomial. In this
chapter, we’ll use the completely factored form of a polynomial to help us
graph it.
The far right and far left of a polyniomial graph
Suppose p(x) = an xn + an−1 xn−1 + an−2 xn−2 + · · · + a0 is a polynomial.
If M is a really big number, then M n is much bigger than M n−1 . (For
example, if M = 1000 then M n is one thousand times bigger than M n−1 .)
In fact, if M is a really, really big number then M n is much bigger than
an−1 M n−1 , or an−2 M n−2 , or an−3 M n−3 , and so on.
Actually, if M is a really really big number, then an M n is much bigger than
the numbers in the previous paragraph, and it even dwarfs their sum:
an−1 M n−1 + an−2 M n−2 + · · · + a0
That means that for really, really big numbers M in the domain of the
polynomial p(x), the size of
p(M ) = an M n + an−1 M n−1 + an−2 M n−2 + · · · + a0
is basically determined by an M n . That’s because while the rest of p(M )
— which is an−1 M n−1 + an−2 M n−2 · · · + a0 — might be large, it is so small
in comparison to an M n that it’s hard to notice it. In other words, while
p(M ) does not equal an M n , it’s hard to tell the difference between the two
in the same way that it would be hard to tell the difference between a bag of
10, 000, 576 pennies and a bag of 10, 002, 073 pennies; both bags have about
10 million pennies.
The end result is that the graph of p(x) looks an awful lot like the graph of
an xn over the part of the x-axis that has the really, really big numbers: the
extreme right portion of the x-axis.
The graph of p(x) also looks an awful lot like the graph of an xn over the
extreme left portion of the x-axis.
Example. Over the far left portion of the x-axis, and over the far right
portion of the x-axis, the graph of q(x) = 27x15 − 2x11 + 3x7 + 6x5 − 4x2 − 8
basically looks like the graph of its leading term: 27x15 . And the graph of
27x15 is the graph of x15 stretched by 27, which is pretty much the same
graph as the graph for x15 .
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The
n
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portion
of
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graph
of
polynomial
p(x)
=
+
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portion of
ofthe
thegraph
graphof
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polynomialp(x)
p(x)=
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+aaan−1
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looks
portion
++······+
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nn
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like
(if
≥
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depends
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the
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It just
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depends on
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leading term,
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What the
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Steps for graphing a completely factored polynomial p(x).
Steps
Steps for
for graphing
graphing aa completely
completely factored
factored polynomial
polynomial p(x).
p(x).
1: The roots of p(x) are the x-intercepts. You can read them off from the
1:
roots
of
the
can
1: The
The
roots
of p(x)
p(x)
are
the x-intercepts.
x-intercepts.
You
can read
read them
them off
off from
from the
the
monic
linear
factors
of aare
completely
factoredYou
polynomial.
monic
linear
factors
of
a
completely
factored
polynomial.
monic
linear factors
of =
a completely
The polynomial
p(x)
−5(x + 3)(xfactored
− 4)(x −polynomial.
4)(x22 + 2x + 6) is completely
The
=
The polynomial
polynomial
p(x)−3
= −5(x
−5(x
+3)(x
3)(x−
−4)(x
4)(x−
−4)(x
4)(x2 +
+ 2x
2x +
+ 6)
6) isis completely
completely
factored.
Its roots p(x)
are
and 4.+
factored.
factored. Its
Its roots
roots are
are −3
−3 and
and 4.
4.
14.
-.3
2: Pick any number on the x-axis between consecutive pairs of x-intercepts.
2:
the
x-axis
between
consecutive
of
2: Pick
Pick
any
number
on
the
x-axiswas
between
consecutive
pairs
of x-intercepts.
x-intercepts.
Let’s
sayany
thenumber
numberon
you
picked
b. If p(b)
> 0, butpairs
a giant
dot directly
Let’s
say
the
number
you
picked
was
b.
If
p(b)
>
0,
but
a
giant
Let’sthe
sayb.the
was
b. If bp(b)
> 0,<but
dot directly
directly
above
Putnumber
a giantyou
dotpicked
directly
below
if p(b)
0. a giant dot
above
the
b.
Put
a
giant
dot
directly
below
b
if
p(b)
<
0.
above
b. Putinabetween
giant dot−3
directly
b ifp(x)
p(b)=<−5(x
0. + 3)(x − 4)(x −
0 is athe
number
and 4,below
and for
00 is
in
between
and
3)(x
−
2is aa number
number
inhave
between
−3−5(3)(−4)(−4)(6)
and 4,
4, and
and for
for p(x)
p(x)
= −5(x
−5(x
+
3)(x
−a4)(x
4)(x
−
4)(x
+ 2x
+ 6) we
p(0) −3
=
<=
0,
so
we+
can
put−
giant
22
4)(x
+
2x
+
6)
we
have
p(0)
=
−5(3)(−4)(−4)(6)
<
0,
so
we
can
put
a
giant
4)(xdirectly
+ 2x +below
6) we 0.
have p(0) = −5(3)(−4)(−4)(6) < 0, so we can put a giant
dot
dot
directly
below
0.
dot directly below 0.
Li-
-3
0
137
143
137
3:
3: The
Thefar
far right
right and
and left
left portion
portion ofof the
the graph
graph ofof p(x)
p(x) looks
looks like
like the
the graph
graph
ofof its
its leading
leading term.
term. Draw
Draw what
what the
the graph
graph ofof the
the leading
leading term
term looks
looks like
like on
on
the
thefar
farright
rightand
andleft
leftsides
sidesofofyour
yourpicture.
picture.
The
Theleading
leadingterm
termofofp(x)
p(x)==−5(x
−5(x++3)(x
3)(x−−4)(x
4)(x−−4)(x
4)(x22++2x
2x++6)
6) isis −5x
−5x55,,
and
andthat
thatlooks
lookslike
likethe
thegraph
graphofofxx55 turned
turnedupside
upside down.
down.
N
‘IL.
-3
0
.
4: Draw
Drawany
anytype
typeofofsmooth,
smooth,curvy,
curvy,and
andcontinuous
continuousline
linethat
thatpasses
passesthrough
through
4:
2
2
all ofof the
the points
points in
in RR that
that you
you labeled
labeled from
from Steps
Steps 11 and
and 2,2, that
that does
does not
not
all
touch the
the x-axis
x-axis at
at any
any points
points not
not listed
listed in
in Step
Step 1,1, and
and that
that meets
meets up
up with
with
touch
thepieces
piecesofofthe
thegraph
graphyou
youdrew
drewin
inStep
Step3.3.
the
LI-
p(x)
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138
144
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22
Problem.
Problem.
The
polynomial
q(x)
7(x+2)(x−3)(x−5)(x
+1)
completely
Problem.The
Thepolynomial
polynomialq(x)
q(x)==
=7(x+2)(x−3)(x−5)(x
7(x+2)(x−3)(x−5)(x2+1)
+1)isis
iscompletely
completely
factored.
Graph
it.
factored.
Graph
it.
factored. Graph it.
Solution.
Solution.
Solution.
1:
1:
The
monic
linear
factors
q(x)
are
(x
2),
(x
3),
and
(x
5).
So
1: The
Themonic
moniclinear
linearfactors
factorsofof
ofq(x)
q(x)are
are(x
(x++
+2),
2),(x
(x−−
−3),
3),and
and(x
(x−−
−5).
5). So
So
the
roots
of
q(x)
are
−2,
3,
and
5.
Draw
these
three
points
on
the
x-axis.
the
the roots
roots of
of q(x)
q(x) are
are −2,
−2, 3,
3, and
and 5.
5. Draw
Draw these
these three
three points
points on
on the
the x-axis.
x-axis.
-2.
3
5
2:
2: Choose
Choose any
any number
number between
between −2
−2 and
and 3,3, for
for example,
example, the
the number
number 00 isis
between
between −2
−2 and
and3.3. Then
Thencheck
checkto
tosee
seeififq(0)
q(0)isispositive
positiveor
ornegative:
negative: q(0)
q(0)==
22
7(0
7(0++2)(0
2)(0−−3)(0
3)(0−−5)(0
5)(0 ++1)
1) == 7(2)(−3)(−5)(1)
7(2)(−3)(−5)(1) isis aa positive
positive number,
number, so
so
draw
a
dot
above
0.
draw a dot above 0.
Similarly,
Similarly,choose
chooseaapoint
pointbetween
between33and
and5,5,say
say4.4. Check
Checkthat
thatq(4)
q(4)==7(4
7(4++
222
2)(4
2)(4−−3)(4
3)(4−−5)(4
5)(4 ++1)
1)==7(6)(1)(−1)(17)
7(6)(1)(−1)(17)isisnegative,
negative,so
sowe
wedraw
drawaadot
dotbelow
below
4.4.
-2.
a
S
145
139
139
5
5
3:
3: The
The leading
leadingterm
termof
ofq(x)
q(x)isis
is7x
7x5.5.. Draw
Drawthe
thepart
partof
of7x
7x55that
thatisisisto
tothe
the
3:
The
leading
term
of
q(x)
7x
Draw
the
part
of
7x
that
to
the
left
of
everything
you’ve
drawn
in
your
picture
so
far.
Draw
the
part
of
the
left of
of everything
everything you’ve
you’ve drawn
drawn in
in your
your picture
picture so
so far.
far. Draw
Draw the
the part
part of
of the
the
left
555
graph
graphof
of7x
7x that
thatisis
isto
tothe
theright
rightof
ofeverything
everythingyou’ve
you’vedrawn
drawnso
sofar.
far.
graph
of
7x
that
to
the
right
of
everything
you’ve
drawn
so
far.
/
4:
4:Draw
Drawthe
thegraph
graphof
ofaaafunction
functionthat
thatconnects
connectseverything
everythingyou’ve
you’vedrawn,
drawn,but
but
4:
Draw
the
graph
of
function
that
connects
everything
you’ve
drawn,
but
make
sure
it
only
touches
the
x-axis
at
the
x-intercepts
that
you’ve
already
make sure
sure it
it only
only touches
touches the
the x-axis
x-axis at
at the
the x-intercepts
x-intercepts that
that you’ve
you’ve already
already
make
labelled.
labelled. That
Thatisis
ismore
moreor
orless
lesswhat
whatthe
thegraph
graphof
ofq(x)
q(x)looks
lookslike.
like. That’s
That’sour
our
labelled.
That
more
or
less
what
the
graph
of
q(x)
looks
like.
That’s
our
answer.
answer.
answer.
-2
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140
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146
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Exercises
For #1-6, graph the given completely factored polynomials.
1.) 4(x − 3)(x − 5)
2.) −(x − 1)(x2 + x + 5)
3.) −6(x + 4)(x + 4)(x − 2)(x − 3)(x2 + 1)(x2 + 3)
4.) 2(x − 3)(x − 3)(x − 3)(x − 6)(x − 6)(x2 + 2x + 7)
5.) −(x − 1)(x − 2)(x − 2)(x − 3)(x2 + x + 5)
6.) 5(x − 4)(x2 − 2x + 4)(x2 + 3x + 5)(x2 + 4)
For #7-10, first completely factor, and then graph, the given polynomials.
7.) 2x2 + 2x − 24
8.) 7x2 − 3x + 4
9.) −x3 + 6x2 + 7x
10.) 3x4 − 9x2 − 12
11.) How can what we know about the far right and left of the graph of an
odd degree polynomial be used to show that any polynomial of odd
degree has at least one root?
147