1
Math 1404 Precalculus Polynomial and Rational Functions 1
Polynomial and Rational
Functions
Math 1404 Precalculus Polynomial and Rational Functions --
Polynomial Functions and Their Graphs
2
Polynomial functions and Their
Graphs
Math 1404 Precalculus Polynomial and Rational Functions --
Polynomial Functions and Their Graphs
3
Polynomial functions and Their
Graphs
A Polynomial of degree n is a function of the form
where n is a nonnegative integer and
• The number are coefficients.
• is the constant coefficient or term.
• is the leading coefficient.
01
1
1)( axaxaxaxP n
n
n
n ++++=−
−L
.0≠na
naaa ,,, 10 K
0a
na
Math 1404 Precalculus Polynomial and Rational Functions --
Polynomial Functions and Their Graphs
4
Examples
Explain why each of the following functions is or is not a polynomial function.
1. f(x) = 2x5- 3x4+ x3- 5x2- x + 3
2. g(z) = (z - ½)2
3. f(x) = 1/x
4. r(x) =
5.
6.
7.
3)( 2++= xxxG
x7)( 3/2
+= xxH
3−= xy
2
Math 1404 Precalculus Polynomial and Rational Functions --
Polynomial Functions and Their Graphs
5
Graphs of Power Functions
xy =2xy =
3xy =
6xy =5xy =
4xy =
Math 1404 Precalculus Polynomial and Rational Functions --
Polynomial Functions and Their Graphs
6
Graphs of Polynomials
3xy =3xy −=
4xy =
23−−= xy 2)3( 3
−+−= xy
4xy −= 34+−= xy 3)2( 4
+−−= xy
Math 1404 Precalculus Polynomial and Rational Functions --
Polynomial Functions and Their Graphs
7
End Behavior of Polynomials
End behavior describes behavior (of y) as x becomes large in either positive or negative direction.
x → −∞ means x becomes large in the negative direction
x → ∞ means x becomes large in the positive direction
– For polynomial with even degree
y → ∞ as x → −∞ and y → ∞ as x → ∞ or
y → −∞ as x → −∞ and y → −∞ as x → ∞
– For polynomial with odd degree
y → ∞ as x → −∞ and y → −∞ as x → ∞ or
y → −∞ as x → −∞ and y → ∞ as x → ∞
Math 1404 Precalculus Polynomial and Rational Functions --
Polynomial Functions and Their Graphs
8
End Behavior of Even Degree
Polynomials
y → ∞ as x → −∞ y → ∞ as x → ∞
y → −∞ as x → −∞ y → −∞ as x → ∞
3
Math 1404 Precalculus Polynomial and Rational Functions --
Polynomial Functions and Their Graphs
9
End Behavior of Odd Degree
Polynomials
y → ∞ as x → −∞
y → −∞ as x → −∞
y → ∞ as x → ∞
y → −∞ as x → ∞
Math 1404 Precalculus Polynomial and Rational Functions --
Polynomial Functions and Their Graphs
10
Using Zeros to Graph Polynomials
If y = P(x) is a polynomial and P(c) = 0, then a
number c is a zero of P.
The following statements are equivalent
c is a zero or root of P.
x = c is a solution of the equation P(x) = 0.
x − c is a factor of P(x)
Math 1404 Precalculus Polynomial and Rational Functions --
Polynomial Functions and Their Graphs
11
Example
Find the zeros for xxxy 623−+=
Math 1404 Precalculus Polynomial and Rational Functions --
Polynomial Functions and Their Graphs
12
Example
Find the zeros for )2)(1)(2( −−+= xxxxy
4
Math 1404 Precalculus Polynomial and Rational Functions --
Polynomial Functions and Their Graphs
13
Intermediate Value Theorem of
Polynomials
If P is a polynomial function and P(a) and P(b) have
opposite signs, then there exists at least one value
c between a and b such that P(c) = 0.
b
a
c
f(a)
f(b)
Math 1404 Precalculus Polynomial and Rational Functions --
Polynomial Functions and Their Graphs
14
Problems on Page 262
Sketch the graph of the function.
12.
25.
27.
)2)(1)(1()( −+−= xxxxP
234 23)( xxxxP +−=
xxxxP 12)( 23++−=
Math 1404 Precalculus Polynomial and Rational Functions --
Polynomial Functions and Their Graphs
15
Local Extrema of Polynomials
f(x) = x3 + 8x2 + 13x − 2
Math 1404 Precalculus Polynomial and Rational Functions --
Polynomial Functions and Their Graphs
16
Local Extrema of Polynomials
If P(x) is a polynomial degree n, then the graph of P
has at most n − 1local extrema (turning points).
5
Math 1404 Precalculus Polynomial and Rational Functions --
Polynomial Functions and Their Graphs
17
Practice Problems on Page 262
5-10, 11, 13, 15, 23, 29, 31, 35, 43-46, 83-84
Math 1404 Precalculus Polynomial and Rational Functions --
Dividing Polynomial
18
Dividing Polynomial
Math 1404 Precalculus Polynomial and Rational Functions --
Dividing Polynomial
19
622 23+−−− xxxx
622
2
23
x
xxxx +−−−
2
622
23
2
23
xx
x
xxxx
−
+−−−
2
23
2
23
2
622
x
xx
x
xxxx
+−
+−−−
xx
xx
xx
xxxx
2
2
622
2
23
2
23
−
+−
+
+−−−
xx
xx
xx
xx
xxxx
2
2
2
622
2
2
23
2
23
−
−
+−
+
+−−−
0
2
2
2
622
2
2
23
2
23
xx
xx
xx
xx
xxxx
+−
−
+−
+
+−−−
60
2
2
2
622
2
2
23
2
23
+
+−
−
+−
+
+−−−
xx
xx
xx
xx
xxxx
Long Division of Polynomials
Problem 2 p. 278
The quotient is x2 + x and the remainder is 6
Math 1404 Precalculus Polynomial and Rational Functions --
Dividing Polynomial
20
Division Algorithm
If P(x) and D(x) are polynomials, D(x) ≠ 0, then there exist unique polynomial Q(x) and R(x) such that
P(x) = D(x) ⋅ Q(x) + R(x)
where R(x) is either 0 or of degree less than the degree of D(x). – P(x) – dividend
– D(x) -- divisor
– Q(x) -- quotient
– R(x) -- remainder
6
Math 1404 Precalculus Polynomial and Rational Functions --
Dividing Polynomial
21
Synthetic Division
Example
17
279
109
186
276
3
96
102793
2
2
23
2
23
−
−
+−
+−
−
+−
−+−−
x
x
xx
xx
xx
xx
xxxx
3
10279 23
−
−+−
x
xxx
179−61
r=17x2− 6x + 9
27−183
−1027−913
Math 1404 Precalculus Polynomial and Rational Functions --
Dividing Polynomial
22
Problem 33 Page 271
Using synthetic division to find the quotient and the
remainder of
2
1
1232 23
−
+−+
x
xxx
Math 1404 Precalculus Polynomial and Rational Functions --
Dividing Polynomial
23
Remainder Theorem
If the polynomial P(x) is divided by x − c, then the remainder
is the value P(c).
Proof: If the divisor is in the form x − c for some real number
c, then the remainder must be a constant (since the degree
of the remainder is less than the degree of the divisor)
Where r is the remainder, then
rrcQcccP =+⋅−= )()()(
rxQcxxP +⋅−= )()()(
Math 1404 Precalculus Polynomial and Rational Functions --
Dividing Polynomial
24
Factor Theorem
c is the zero of a polynomial P(x) iff (if and only if) x − c is a factor of P(x).
Proof:
1. If c is a zero of P(x), that is P(c) = 0, then by the remainder theorem
This implies x − c is a factor of P(x).
2. Let x − c be a factor of P(x), then
Therefore
Complete Factorization Theorem
)()()( xQcxxP ⋅−=
0)()()( +⋅−= xQcxxP
0)()()( =⋅−= cQcccP
7
Math 1404 Precalculus Polynomial and Rational Functions --
Dividing Polynomial
25
Problems on Page 271
Find a polynomial of the specified degree that has
the given zeros
57. Degree 3; zeros: −1, 1, 3
58. Degree 4; zeros: −2, 0, 2, 4
Math 1404 Precalculus Polynomial and Rational Functions --
Dividing Polynomial
26
Practice Problems on Page 270
23,24,27,28,31,35,36,39,40,43,44,51,52,55,56,59,60,
63-66,67.
Math 1404 Precalculus Polynomial and Rational Functions --
Real Zeros of Polynomial
27
Real Zeros of Polynomials
Math 1404 Precalculus Polynomial and Rational Functions --
Real Zeros of Polynomial
28
Rational Zeros Theorem
The Polynomial has integer coefficients, then every rational zero of P is of the form p/q where p is a factor of the constant coefficient a0 and q is a factor of the leading coefficient an.
Proof: If p/q is a rational zero, in lowest terms, then from Factor Theorem
Since p is the factor of the left hand side, p is also the factor of the right hand side. Also, since p/q is in lowest terms, p and q have no factor in common, therefore p must be a factor of a0.
01
1
1)( axaxaxaxP n
n
n
n ++++=−
−L
( ) nnn
n
n
n
nnn
n
n
n
n
n
n
n
qaqaqpapap
qapqaqpapa
aq
pa
q
pa
q
pa
q
pP
0
1
1
2
1
1
0
1
1
1
1
01
1
1
0
0
−=+++⇒
=++++⇒
=+
++
+
=
−−
−
−
−−
−
−
−
L
L
L
8
Math 1404 Precalculus Polynomial and Rational Functions --
Real Zeros of Polynomial
29
Problem 4 on Page 279
List all possible rational zeros given by the Rational Zeros Theorem.
4.
Possible rational zeros are ±1/1, ±2/1, ±3/1, ±4/1, ±6/1, ±12/1, ±1/2, ±3/2, ±1/3, ±2/3, ±4/3, ±6/1
±1/1, ±2/1, ±3/1, ±4/1, ±6/1, ±12/1,
±1/2, ±2/2, ±3/2, ±4/2, ±6/2, ±12/2,
±1/3, ±2/3, ±3/3, ±4/3, ±6/3, ±12/3,
±1/6, ±2/6, ±3/6, ±4/6, ±6/6, ±12/6,
1226)( 24++−= xxxxP
Math 1404 Precalculus Polynomial and Rational Functions --
Real Zeros of Polynomial
30
Problem 6 on Page 279
Find all rational zeros of the polynomial.
6.
Possible rational zeros are ±1/1, ±2/1, ±4/1, ±8/1
The zeros are 1, 2, 4
8147)( 23−+−= xxxxP
08−61
x2− 6x + 8 = (x − 2) (x − 4)
8−61
−814−711
Math 1404 Precalculus Polynomial and Rational Functions --
Real Zeros of Polynomial
31
Problem 28 on Page 280
Find all rational zeros of the polynomial.
36.
Possible rational zeros are ±1/1, ±2/1, ±1/2, ±1/3,
±2/3
The zeros are −1, 2, 1/2 −1/3
231276)( 234++−−= xxxxxP
−2 −212
021−1362
0
−1
3
−1−16
6x2− x − 1 = (2x − 1) (3x + 1)
−213−6
2−12−76−1
Math 1404 Precalculus Polynomial and Rational Functions --
Real Zeros of Polynomial
32
Descartes’ Rule of Signs
Let P be a polynomial with real coefficients.
1. The number of positive real zeros of P(x) is
either equal to the number of variations in sign in
P(x) or is less than by an even whole number.
2. The number of negative real zeros of P(x) is
either equal to the number of variations in sign in
P(−x) or is less than that by an even whole
number.
9
Math 1404 Precalculus Polynomial and Rational Functions --
Real Zeros of Polynomial
33
Descartes’ Rule of signs and
Upper and Lower Bounds for Roots
2 or 0
1
3 or 1
+ Real Zeros
00P(−x) = − x5 − 4x3− x2 − 6x
2P(x) = x5 + 4x3− x2 + 6x
2 or 02P(−x) = −x3 − x2 + x − 3
1P(x) = x3− x2 − x − 3
00P(−x) = −x3− 7x2 − 14x − 8
3P(x) = x3− 7x2 + 14x− 8
− Real ZerosVariations in signPolynomial
Math 1404 Precalculus Polynomial and Rational Functions --
Real Zeros of Polynomial
34
The Upper and Lower bounds
Theorem
Let P be a polynomial with real coefficients.
1. If we divide P(x) by x − b (with b > 0) using synthetic division, and if the row that contains the quotient and remainder has no negative entry, then b is an upper bound for the real zeros of P.
2. If we divide P(x) by x − a (with a < 0) using synthetic division, and if the row that contains the quotient and remainder has entries that are alternately non-positive and negative, then a is an lower bound for the real zeros of P.
Math 1404 Precalculus Polynomial and Rational Functions --
Real Zeros of Polynomial
35
Problem 59 on Page 280
Show that the given value for a and b are lower and
upper bound for the real zeros of the polynomial.
67. 2,3 ;939108)( 23=−=+−+= baxxxxP
3513268
26
9
5216
−391082
03−148
−9
9
42−24
−39108−3
Math 1404 Precalculus Polynomial and Rational Functions --
Real Zeros of Polynomial
36
Problem 60 on Page 280
Show that the given value for a and b are lower and
upper bound for the real zeros of the polynomial.
68. 6,0 ;1924173)( 234==+−+−= baxxxxxP
1−924−173
0
−9
000
124−1730
11713013
180
−9
1026618
124−1736
10
Math 1404 Precalculus Polynomial and Rational Functions --
Real Zeros of Polynomial
37
Using Algebra and
Graphing Devices to
Solve Polynomial Equations032743 234
=−−−+ xxxx
Math 1404 Precalculus Polynomial and Rational Functions --
Real Zeros of Polynomial
38
Using Graphing Devices to
Solve Inequalities (51 p. 110)
6611 23+≤+ xxx 06116 23
≤−+−⇒ xxx
Math 1404 Precalculus Polynomial and Rational Functions --
Real Zeros of Polynomial
39
Using Graphing Devices to
Solve Inequalities (52 p. 110)
192416 23−−>+ xxx 0192416 23
>+++⇒ xxx
Math 1404 Precalculus Polynomial and Rational Functions --
Real Zeros of Polynomial
40
Practice Problems on Page 279
2, 3, 7-10, 11, 15, 16, 19, 24, 27, 32, 37, 41, 44, 49,
50, 51, 53, 56, 60, 62, 65, 66, 83, 85, 100.
11
Math 1404 Precalculus Polynomial and Rational Functions --
Complex Numbers
41
Complex Numbers
Math 1404 Precalculus Polynomial and Rational Functions --
Complex Numbers
42
Definition of Complex Numbers
A complex number is an expression of the form
a + bi
where a and b are real number and i2 = −1.
– The real part is a.
– The imaginary part is b.
Two complex number are equal if and only if the
real and the imaginary part are equal.
Math 1404 Precalculus Polynomial and Rational Functions --
Complex Numbers
43
Problems on Page 289
2. −6 + 4i
3. 3
52 i−−
Math 1404 Precalculus Polynomial and Rational Functions --
Complex Numbers
44
Arithmetic of Complex Numbers
• Addition:
(a + bi) + (c + di) = (a + c) + (b + d)i
• Subtraction:
(a + bi) − (c + di) = (a − c) + (b − d)i
• Multiplication:
(a + bi)(c + di) = (ac − bd) + (ad + bc)i
• Division:
(c + d)2
(ac + bd) + (bc − ad)i
(c + di) (c − di)
(a + bi) (c − di)=
c + di=
a + bi
12
Math 1404 Precalculus Polynomial and Rational Functions --
Complex Numbers
45
Square Root of Negative Numbers
If −r is negative, then the principle square root of −r
is
The two square root of −r are and
rir =−
.ri−ri
Math 1404 Precalculus Polynomial and Rational Functions --
Complex Numbers
46
Powers of i
1
1
1
1
8
7
6
5
4
3
2
=
−=
−=
=
=
−=
−=
=
i
ii
i
ii
i
ii
i
ii
=
−=
−=
=
=
1 then4
then3
1 then2
then1
4/ ofremainder theIf
n
n
n
n
i
ii
i
ii
n
Math 1404 Precalculus Polynomial and Rational Functions --
Complex Numbers
47
Problems on Page 289-290
14. (3 − 2i) + (−5 − )
18. (−4 + i) − (2 − 5i)
26. (5 − 3i)(1 + i)
34.
45.
53.
i
i
43
5
+
−
i3
1
100i
21
82
−+
−+
Math 1404 Precalculus Polynomial and Rational Functions --
Complex Numbers
48
Complex Roots of Quadratic
functions
64. 2x2 + 3 = 2x
13
Math 1404 Precalculus Polynomial and Rational Functions --
Complex Numbers
49
Practice Problems on Page 289
1,4,5,8,9,13,20,25,30,33,36,41,46,49,52,57,62,65,70.
Math 1404 Precalculus Polynomial and Rational Functions --
Complex Zero and the Fundamental
Theorem of Algebra
50
Complex Zeros and the Fundamental
Theorem of Algebra
Math 1404 Precalculus Polynomial and Rational Functions --
Complex Zero and the Fundamental
Theorem of Algebra
51
Fundamental Theorem of Algebra
Every polynomial
with complex coefficients has at least one complex
zero.
)0,1( )( 01
1
1 ≠≥++++=−
− n
n
n
n
n anaxaxaxaxP L
Math 1404 Precalculus Polynomial and Rational Functions --
Complex Zero and the Fundamental
Theorem of Algebra
52
Complete Factorization Theorem
If P(x) is a polynomial of degree n > 0, then there exist complex numbers
such that
Proof: By the Fundamental Theorem of Algebra, P(x) has at least one zero, c1. By the Factor Theorem, P(x) can be factored as
where Q1(x) is of degree n − 1.
Similarly, Q1(x) has at least one zero, c2 and where Q2(x) is of degree n − 2. Continuing the process for n steps, we obtain a final quotient Qn(x) is of degree 0 as a constant number a. Therefore
Zeros Theorem
)0 with( ,,,, 21 ≠accca nK
)())(()( 21 ncxcxcxaxP −−−= L
)()()( 11 xQcxxP −=
)()()( 22 xQcxxP −=
)())(()( 21 ncxcxcxaxP −−−= L
14
Math 1404 Precalculus Polynomial and Rational Functions --
Complex Zero and the Fundamental
Theorem of Algebra
53
Zeros and Their Multiplicities
Problems on page 298:
Factor the polynomial completely and find all its
zeros. State multiplicity of each zero.
14.
19.
30.
94)( 2+= xxP
1)( 4−= xxP
6416)( 36++= xxxP
Math 1404 Precalculus Polynomial and Rational Functions --
Complex Zero and the Fundamental
Theorem of Algebra
54
Zeros Theorem
Every polynomial of degree n ≥ 1 has exactly n zeros,
provided that a zero of multiplicity k is counted k times.
Proof: Let P be a polynomial of degree n. By Complete
Factorization Theorem
Suppose c is a zero of P other than
Then and therefore P has exactly n zeros.
)())(()( 21 ncxcxcxaxP −−−= L
0)())(()( 21 =−−−= nccccccacP L
nccc ,,, 21 K
ncccc ,,, 21 K=
Math 1404 Precalculus Polynomial and Rational Functions --
Complex Zero and the Fundamental
Theorem of Algebra
55
Conjugate Zeros Theorem
If the polynomial P has real coefficients, and if the complex number z is a zero of P, then its complex conjugate is also a zero of P.
Proof: Let where each coefficient is real. Suppose that P(z) = 0. We must prove that
z
01
1
1)( axaxaxaxP n
n
n
n ++++=−
−L
.0)( =zP
00)(
)(
01
1
1
01
1
1
01
1
1
01
1
1
===
++++=
++++=
++++=
++++=
−
−
−
−
−
−
−
−
zP
azazaza
azazaza
azazaza
azazazazP
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
L
L
L
L
Math 1404 Precalculus Polynomial and Rational Functions --
Complex Zero and the Fundamental
Theorem of Algebra
56
Problems on Page 298
Find a polynomial with integer coefficients that
satisfies the given conditions.
33. Q has degree 3, and zeros 3, 2i, and −2i
37. R has degree 4, and zeros 1 − 2i, and 1, with 1 a
zero of multiplicity 2.
15
Math 1404 Precalculus Polynomial and Rational Functions --
Complex Zero and the Fundamental
Theorem of Algebra
57
Linear and Quadratic Factors
TheoremEvery polynomial with real coefficients can be factored into a product of linear and
irreducible quadratic factors with real coefficients.
Proof: If c = a + bi, then has real coefficients.
So, if P is a polynomial with real coefficients, then by the Complete Factorization Theorem
And since the complex roots occur in conjugate pairs, then P can be factored into a product of linear and irreducible quadratic factors with real coefficients.
)(2
)()(
)]))][()[(
)]()][([))((
222
22
baaxx
biax
biaxbiax
biaxbiaxcxcx
++−=
−−=
−−+−=
−−+−=−−
))(( cxcx −−
)())(()( 21 ncxcxcxaxP −−−= L
Math 1404 Precalculus Polynomial and Rational Functions --
Complex Zero and the Fundamental
Theorem of Algebra
58
Problems on Page 299
Find all solutions of the polynomial equation by
factoring and using the quadratic formula.
48.
50.
9982)( 23−+−= xxxxP
3222)( 234−−−−= xxxxxP
Math 1404 Precalculus Polynomial and Rational Functions --
Complex Zero and the Fundamental
Theorem of Algebra
59
Practice Problems on Page 298
3,5,8,15,23,28,29,31,36,41,44,49,55,58,70.
Math 1404 Precalculus Polynomial and Rational Functions --
Rational Functions
60
Rational Functions
16
Math 1404 Precalculus Polynomial and Rational Functions --
Rational Functions
61
Rational Functions and Asymptotes
A rational function is a function of the form
)(
)()(
xQ
xPxr =
Math 1404 Precalculus Polynomial and Rational Functions --
Rational Functions
62
Graph of Rational Function y = 1/x
)(
)()(
xQ
xPxr =
y goes to positive infinityy → ∞
y goes to negative infinityy → −∞
x approaches a from the rightx → a+
x approaches a from the leftx → a−
MeaningSymbol
Math 1404 Precalculus Polynomial and Rational Functions --
Rational Functions
63
Definition of Asymptotes
1. The line x = a is a vertical asymptote of the
function y = f(x) if
y → ∞ as x → a+ or x → a−
y → −∞ as x → a+ or x → a−
2. The line y = b is a horizontal asymptote of the
function y = f(x) if
y → b as x → ∞ or x → −∞
Math 1404 Precalculus Polynomial and Rational Functions --
Rational Functions
64
Transformations of y = 1/x
The graph of rational function of the form
is the graph of shifted, stretched, and or
reflected.
dcx
baxxr
+
+=)(
xy
1=
17
Math 1404 Precalculus Polynomial and Rational Functions --
Rational Functions
65
Problems on Page 313
27. 1
3)(
+=x
xs )1(3)( then,1
)( if +== xrxsx
xr
Math 1404 Precalculus Polynomial and Rational Functions --
Rational Functions
66
Problems on Page 313
28. 2
2)(
−
−=x
xs )2(2)( then,1
)( if −−== xrxsx
xr
Math 1404 Precalculus Polynomial and Rational Functions --
Rational Functions
67
Problems on Page 313
30.2
33)(
+
−=x
xxr
3)2(9)(
2
93)(
then,1
)( if
++−=
+
−+=
=
xsxr
xxr
xxs
Math 1404 Precalculus Polynomial and Rational Functions --
Rational Functions
68
Graphing Rational Functions
1. Factor the numerator and denominator.
2. Find x- and y- intercepts
3. Find vertical asymptotes
4. Find horizontal asymptotes
5. Sketch the graph
18
Math 1404 Precalculus Polynomial and Rational Functions --
Rational Functions
69
Example
53
2)(
+=
x
xxs
0
02
053
2
intercept
=⇒
=⇒
=+
⇒
−
x
x
x
x
x
0503
02intercept =
+⋅
⋅=−y
−∞→−→
∞→−→
−=
+
−
yx
yx
,3
5 as
,3
5 as
3
5 asymptote vertical
3
2 , as
3
2 , as
53
2
53
2
)(
asymptote horizontal
→−∞→
→∞→
+
=
+
=
yx
yx
xxx
xx
x
xs
Math 1404 Precalculus Polynomial and Rational Functions --
Rational Functions
70
Problem 21 on Page 313
65
26)(
2−+
−=
xx
xxr
3
1
026
065
26
intercept
2
=⇒
=−⇒
=−+
−⇒
−
x
x
xx
x
x
3
1
6050
206intercept
2=
−⋅+
−⋅=−y −∞→→∞→→
−∞→−→∞→−→
−=
+−
+−
yxyx
yxyx
,1 as and ,1 as
,6 as and ,6 as
6,1 asymptote vertical
0 , as
0 , as
651
26
65
26
)(
asymptote horizontal
2
2
222
2
22
→−∞→
→∞→
−+
−
=
−+
−
=
yx
yx
xx
xx
xx
x
x
xxx
x
xs
Math 1404 Precalculus Polynomial and Rational Functions --
Rational Functions
71
Problem 53 on Page 313
intercept no
2
063
032
63
intercept
2
2
2
−⇒
−±=⇒
=+⇒
=−−
+⇒
−
x
x
x
xx
x
x
23
6
3020
603intercept
2
2
−=−
=−⋅−
+⋅=−y ∞→→−∞→→
−∞→−→∞→−→
−=
+−
+−
yxyx
yxyx
,3 as and ,3 as
,1 as and ,1 as
1,3 asymptote vertical
3 , as
3 , as
321
63
32
63
)(
asymptote horizontal
2
2
222
2
22
2
→−∞→
→∞→
−−
+
=
−−
+
=
yx
yx
xx
x
xx
x
x
xxx
x
xs
32
63)(
2
2
−−
+=
xx
xxr
Math 1404 Precalculus Polynomial and Rational Functions --
Rational Functions
72
Asymptotes of Rational Functions
Let r be the rational function of the form
1. The vertical asymptotes of r are the lines x = a, where a
is zero of the denominator.
2. (a) if n < m, then r has horizontal asymptote
(b) if n = m, then r has horizontal asymptote
(c) if n > m, then r has no horizontal asymptote.
)(01
1
1
01
1
1
bxbxbxb
axaxaxaxr
m
m
m
m
n
n
n
n
++++
++++=
−
−
−
−
L
L
.m
n
b
ay =
.0=y
19
Math 1404 Precalculus Polynomial and Rational Functions --
Rational Functions
73
Practice Problems on Page 313
5,8,10,11,13,15,16,17,18,23,24,25,29,31,33,35,41,42
,47,75b,77bc,83a.