polynomial and rational functions-chapter...

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



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



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



5

Graphs of Power Functions

xy =2xy =

3xy =

6xy =5xy =

4xy =



6

Graphs of Polynomials

3xy =3xy −=

4xy =

23−−= xy 2)3( 3

−+−= xy

4xy −= 34+−= xy 3)2( 4

+−−= xy



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 → ∞



8

End Behavior of Even Degree

Polynomials

y → ∞ as x → −∞ y → ∞ as x → ∞

y → −∞ as x → −∞ y → −∞ as x → ∞

3



9

End Behavior of Odd Degree

Polynomials

y → ∞ as x → −∞

y → −∞ as x → −∞

y → ∞ as x → ∞

y → −∞ as x → ∞



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)



11

Example

Find the zeros for xxxy 623−+=



12

Example

Find the zeros for )2)(1)(2( −−+= xxxxy

4



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)



14

Problems on Page 262

Sketch the graph of the function.

12.

25.

27.

)2)(1)(1()( −+−= xxxxP

234 23)( xxxxP +−=

xxxxP 12)( 23++−=



15

Local Extrema of Polynomials

f(x) = x3 + 8x2 + 13x − 2



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



17

Practice Problems on Page 262

5-10, 11, 13, 15, 23, 29, 31, 35, 43-46, 83-84


Dividing Polynomial

18

Dividing Polynomial


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


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


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


Dividing Polynomial

22

Problem 33 Page 271

Using synthetic division to find the quotient and the

remainder of

2

1

1232 23

−

+−+

x

xxx


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 +⋅−= )()()(


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


Dividing Polynomial

25


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


Dividing Polynomial

26


23,24,27,28,31,35,36,39,40,43,44,51,52,55,56,59,60,

63-66,67.


Real Zeros of Polynomial

27

Real Zeros of Polynomials



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



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



30


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



31


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



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



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



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.



35


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



36


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



37

Using Algebra and

Graphing Devices to

Solve Polynomial Equations032743 234

=−−−+ xxxx



38

Using Graphing Devices to

Solve Inequalities (51 p. 110)

6611 23+≤+ xxx 06116 23

≤−+−⇒ xxx



39

Using Graphing Devices to

Solve Inequalities (52 p. 110)

192416 23−−>+ xxx 0192416 23

>+++⇒ xxx



40


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


Complex Numbers

41

Complex Numbers


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.


Complex Numbers

43


2. −6 + 4i

3. 3

52 i−−


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


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


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


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

−+

−+


Complex Numbers

48

Complex Roots of Quadratic

functions

64. 2x2 + 3 = 2x

13


Complex Numbers

49


1,4,5,8,9,13,20,25,30,33,36,41,46,49,52,57,62,65,70.


Complex Zero and the Fundamental

Theorem of Algebra

50

Complex Zeros and the Fundamental

Theorem of Algebra



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



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



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



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=



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



Theorem of Algebra

56


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



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



Theorem of Algebra

58


Find all solutions of the polynomial equation by

factoring and using the quadratic formula.

48.

50.

9982)( 23−+−= xxxxP

3222)( 234−−−−= xxxxxP



Theorem of Algebra

59


3,5,8,15,23,28,29,31,36,41,44,49,55,58,70.


Rational Functions

60

Rational Functions

16


Rational Functions

61

Rational Functions and Asymptotes

A rational function is a function of the form

)(

)()(

xQ

xPxr =


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


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 → −∞


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


Rational Functions

65


27. 1

3)(

+=x

xs )1(3)( then,1

)( if +== xrxsx

xr


Rational Functions

66


28. 2

2)(

−

−=x

xs )2(2)( then,1

)( if −−== xrxsx

xr


Rational Functions

67


30.2

33)(

+

−=x

xxr

3)2(9)(

2

93)(

then,1

)( if

++−=

+

−+=

=

xsxr

xxr

xxs


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


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


Rational Functions

70


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

)(


2

2

222

2

22

→−∞→

→∞→

−+

−

=

−+

−

=

yx

yx

xx

xx

xx

x

x

xxx

x

xs


Rational Functions

71


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

)(


2

2

222

2

22

2

→−∞→

→∞→

−−

+

=

−−

+

=

yx

yx

xx

x

xx

x

x

xxx

x

xs

32

63)(

2

2

−−

+=

xx

xxr


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


Rational Functions

73


5,8,10,11,13,15,16,17,18,23,24,25,29,31,33,35,41,42

,47,75b,77bc,83a.

polynomial and rational functions-chapter...

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