7/27/2019 Chapter 3 Polynomials S

1/62

1

7/27/2019 Chapter 3 Polynomials S

2/62

6.1 POLYNOMIALS 6.2 REMAINDER THEOREM,

FACTOR THEOREM AND ZEROS

OF POLYNOMIAL

6.3 PARTIAL FRACTIONS

2

7/27/2019 Chapter 3 Polynomials S

3/62

LEARNING OUTCOMES

Determine the degrees and coefficients of

polynomials.

Carry out elementary operations on polynomials.

Use the remainder and factor theorems.

Find the roots and zeros of a polynomial.

Express rational functions in partial fractions.

3

7/27/2019 Chapter 3 Polynomials S

4/62

3.1 POLYNOMIALS

4

7/27/2019 Chapter 3 Polynomials S

5/62

5

7/27/2019 Chapter 3 Polynomials S

6/62

Polynomial Specialname Degree Leadingcoefficient(a)

(b)

(c)4

(d)

(e)

91242

xx

1053x

xx 924

79 x

Quadratic

Cubic

4

Constant

5

-2

9

4

2

4

1

Quartic

Linear

3

0

Ex. 3.1:

6

7/27/2019 Chapter 3 Polynomials S

7/62

Equality of Two Polynomials and Operations on

Polynomial

Two polynomial are equal if coefficients of similar

terms are the same.

7

7/27/2019 Chapter 3 Polynomials S

8/62

Two polynomials can be added, subtracted and

multiplied.

8

7/27/2019 Chapter 3 Polynomials S

9/629

7/27/2019 Chapter 3 Polynomials S

10/62

10

7/27/2019 Chapter 3 Polynomials S

11/62

The quotient of polynomials will also be consideredwhere the process of division of polynomials by long

division will be discussed.

where the degree ofR(x) is less than the degree ofD(x).

The polynomial Q(x) is called the quotient and R(x) the remainder.11

7/27/2019 Chapter 3 Polynomials S

12/62

35841223

xxxx

22x

2324 xx

12

7/27/2019 Chapter 3 Polynomials S

13/62

35841223

xxxx

2324 xx

xx 362

xx 56 2

xx 322

13

7/27/2019 Chapter 3 Polynomials S

14/62

35841223

xxxx

23 24 xx

xx 362

xx 562

32 x

12 x

2

1322

xx

14

7/27/2019 Chapter 3 Polynomials S

15/6215

7/27/2019 Chapter 3 Polynomials S

16/62

7/27/2019 Chapter 3 Polynomials S

17/62

REMAINDER THEOREM

The remainder theorem offers a method of finding

the remainder without going through the process of

division.

TheoremTHE REMAINDER THEOREM

17

7/27/2019 Chapter 3 Polynomials S

18/6218

7/27/2019 Chapter 3 Polynomials S

19/6219

7/27/2019 Chapter 3 Polynomials S

20/62

20

7/27/2019 Chapter 3 Polynomials S

21/62

21

7/27/2019 Chapter 3 Polynomials S

22/62

22

7/27/2019 Chapter 3 Polynomials S

23/62

23

7/27/2019 Chapter 3 Polynomials S

24/62

24

7/27/2019 Chapter 3 Polynomials S

25/62

25

7/27/2019 Chapter 3 Polynomials S

26/62

26

7/27/2019 Chapter 3 Polynomials S

27/62

TheoremTHE FACTOR THEOREM

27

7/27/2019 Chapter 3 Polynomials S

28/62

DefinitionZEROS OF POLYNOMIAL

28

7/27/2019 Chapter 3 Polynomials S

29/62

Solution

29

7/27/2019 Chapter 3 Polynomials S

30/62

30

7/27/2019 Chapter 3 Polynomials S

31/62

31

7/27/2019 Chapter 3 Polynomials S

32/62

32

7/27/2019 Chapter 3 Polynomials S

33/62

33

7/27/2019 Chapter 3 Polynomials S

34/62

34

7/27/2019 Chapter 3 Polynomials S

35/62

35

7/27/2019 Chapter 3 Polynomials S

36/62

36

7/27/2019 Chapter 3 Polynomials S

37/62

37

7/27/2019 Chapter 3 Polynomials S

38/62

3.3 PARTIAL

FRACTIONS

38

7/27/2019 Chapter 3 Polynomials S

39/62

PROPER FRACTION

Ratio of two polynomials when the degree of the

numerator is less than the degree of the denominator.

IMPROPER FRACTION

Ratio of two polynomials when the degree of the

numerator is greaterthan the degree of the

denominator.

39

7/27/2019 Chapter 3 Polynomials S

40/62

Two simple algebraic fractions can be combined to

become a single compound fraction:

40

7/27/2019 Chapter 3 Polynomials S

41/62

There are some rules to express fractional polynomial

functions as a sum of partial fractions:

41

7/27/2019 Chapter 3 Polynomials S

42/62

42

7/27/2019 Chapter 3 Polynomials S

43/62

43

7/27/2019 Chapter 3 Polynomials S

44/62

44

7/27/2019 Chapter 3 Polynomials S

45/62

45

7/27/2019 Chapter 3 Polynomials S

46/62

46

7/27/2019 Chapter 3 Polynomials S

47/62

47

7/27/2019 Chapter 3 Polynomials S

48/62

48

7/27/2019 Chapter 3 Polynomials S

49/62

49

7/27/2019 Chapter 3 Polynomials S

50/62

50

7/27/2019 Chapter 3 Polynomials S

51/62

51

7/27/2019 Chapter 3 Polynomials S

52/62

52

7/27/2019 Chapter 3 Polynomials S

53/62

53

7/27/2019 Chapter 3 Polynomials S

54/62

54

7/27/2019 Chapter 3 Polynomials S

55/62

55

7/27/2019 Chapter 3 Polynomials S

56/62

56

7/27/2019 Chapter 3 Polynomials S

57/62

57

7/27/2019 Chapter 3 Polynomials S

58/62

58

7/27/2019 Chapter 3 Polynomials S

59/62

e) Degree of the numerator is the same or higher

than the denominator

Long division is carried out so that the degree of

the numerator becomes less than thedenominator.

59

7/27/2019 Chapter 3 Polynomials S

60/62

60

7/27/2019 Chapter 3 Polynomials S

61/62

61

7/27/2019 Chapter 3 Polynomials S

62/62