1.4 polynomials

Chapter 1.4

POLYNOMIAL

Expressions

1

represented only by letters

and whose values may be

arbitrarily chosen depending

on the situation.

Types of Quantities

Variables

2

a quantity whose value is fixed

and may not be changed during

a particular discussion.

Types of Quantities

Constant

3

Example 1.4.1

Consider the formula

- force

- mass

- acceleration

, , and are variables.

F ma

F

m

a

F m a4

Example 1.4.1

In the formula for computing the

circumference of a circle

and are constants,

while is a varia

2

2

ble.

r

r

5

Algebraic Expressions

Any combination of numbers and

symbols related by the operations

from the previous sections will be

algebraic excalle pressd an ion.

6

2 2

3 10

2 3

2

are algebraic expressions

x y

xx y xy x

y

x y

x y

Example 1.4.2

7


2

Each part of the expression separated by + or ,

together with its sign, is called a of the

algebraic expression

te

.

2 3 2 5

rm

x xy xy

8

Types of Algebraic

Expressions

An algebraic expression consisting of

1 term - monomial

2 terms - binomial

3 terms - trinomial

any number of terms - multinomial

9


Each factor of a term may be call

coefficient

ed

the of the others.

4

4

4

4

4

In 3 ,

3 is the coefficient of .

3 is the coefficient of .

3 is the coef

numerical coefficien

ficient of .

3 is the .

is the

t

literal coefficient.

u v

u v

u v

v u

u v 10


2 2

Terms having the same literal

coefficient similar

terms

s are called

.

3 2 5x xy x y

11

Polynomial Expressions

A is an algebraic

expression involving only non-

negative integra

polynomi

l powers

al in

of .

t

t

12

Example 1.4.3

2 3Is 2 3 1 a polynomial in ?

Is it a polynomial in ?

Is it a polynomial in ?

xt yt t

x

y

13

Polynomials

2

2 2

A polynomial in can be expressed

as .

2 1

A polynomial in and can be

expressed , as .

,

P t

Q

t

P t t t

x y

Q x y x

x

y

y

14

Polynomials

The notations can be used to

find the value of the expression

when the value of the variable

is given.

15

Example 1.4.4

1If , then

2

14 4 2

2

10 0 0

2

S t gt

S g g

S g

16

Example 1.4.4

If 7, then

1 7

7

is constant polynomial a .

C x

C

C

C

17

Polynomials

The of a term of a polynomial

is the exponent of the

variable.

The of a

degree

in a variable

degree in

some variab

term of a polynomial

is the sum of the degrees

of all variables in that t

les

erm.

18

Example 1.4.5

2

3 4

4 10

The degree of in is 2.


The degree of in and is 14.

x y x

x y y

x y x y

19

Polynomials

The degree of the polynomial in some

variables is the de highestgree of its term of

degree in those variables.

20

Example 1.4.6

3

4 3 2 2 3

1. 3 2


2. , 3 2



The degree of in and is 4.

P x x x

P x

Q x y x x y x y y

Q x

Q y

Q x y

21

Addition of Polynomials

To add polynomials, w add

similar t

e

erms.

3 2

3 2

3 2

If 2 1 and

3 5 10

2 5 9

P x x x

Q x x x x

P x Q x x x x

22

Example 1.4.7

Add the following:

5

2 4 3

2

1. 2 3 7

3 5

P x x x x

Q x x x x

P x Q x 5x 4x 33x 2x 3x 2

3 2 2 3

3 2 2 3

2. , 2 3 4 7 5

, 6 7 10

P x y x x y xy y

Q x y y xy x y x23

Multiplication of Polynomials

distributive property

To multiply two polynomials, we

apply the ,

the , and laws of exponents add

similar terms.

24

Example 1.4.8

2 2 2

3 2

3 2

3 2

Perform the indicated operations.

1. 2 5 3 2 2 3 2 5 3 2

6 4 15 10

6 4 15 10

6 15 4 10

t t t t t

t t t

t t t

t t t

25

Example 1.4.8

2 2

3 4 2 2 2 2

3 4 2 2 2

2. 2 3 2 4

6 4 8 3 2 4

6 4 10 3 4

x y xy x y

x y x x y xy x y y

x y x x y xy y

26

Special Products

2 2

Binomial Sum and Difference

x y x y x y

27

Example 1.4.9

2

3 3 2 6

2 3 2 3 4 2 6

Perform the indicated operation.

Use special products.

1. 2 2 4

2. 3 3 9

3. 5 2 5 2 25 4

x x x

x y x y x y

a b c a b c a b c

2 2x y x y x y

28

Special Products

2 2 2

Square of a Binomial

2x y x xy y

29

Example 1.4.10

2 2

2 2 2

22 4 2 2 2

Perform the indicated operation.

Use special products.

1. 3 6 9

2. 2 3 4 12 9

3. 5 10 25

x x x

a b a ab b

p q r p q p qr r

2 2 22x y x xy y

30

Special Products

3 3 2 2 3

3 3 2 2 3

3

Cube

3

3 3

of a Binomial

x y x x y y y

x y x x y xy y

31

Example 1.4.11

3 3

3 3 2

3

3 2 2 3

2 2 3

3 3 2 2 3

1. 2 6 12 8

2. 2 3

8

3

36 5

3

3 3

27

x x x x

a b

a

x y x x

a b ab

y y y

x y x x y xy

b

y

32

Special Products

2 2 3 3

2 2 3 3

Binomial and Trinomial

x y x xy y x y

x y x xy y x y

33

Example 1.4.12

2 2

2 3

2

3 3

2 2 3 3

3

1. 3 3 9 27

2. 3 2 9 6 4 27 8

x

x y x xy y x y

x y

x x x

y y

x xy y x y

y y

34

Factoring

pr

A p

odu

olynom

ct

of t

facto

wo or

ial is

more

if it can be

polynomials w

ra

it

ble

h

int

e

expressed

ger coeffi

as

cie

a

in

nts.

35

Factoring

A factorization is if each

factor

complete

prime facis a tor.

36

Special Products

2 2

2 2 2

2 2 3 3

2

x y x y x y

x y x xy y

x y x xy y x y

37

Example 1.4.13

4 4

2 24 4 2 2

2 2 2 2

2 2

Factor completely.x y

x y x y

x y x y

x y x y x y

38

Example 1.4.13

2 2

4 4 2 2

But these violate the restrictions

hence,

is a complete factorization.

x y x yi x yi

x y x y x y

x y x y x y x y

39

Example 1.4.14

2 2 3 3 4 5 2 2 3

3 4 6 2 4 6

2 22 3

2 3 2 3

Factor the following polynomials

completely.

1. 2 1 2

2. 25 25

5

5 5

x y x y x y x y xy xy

x y xd x x y d

x xy d

x xy d xy d

40

Example 1.4.14

22

22

2 2 2 2

2

3. 8 16 4

4. 4 12 9 2 3

5. 5 30 45 5 6 9

5 3

x x x

y y y

y yz z y yz z

y z

2 2 22x y x xy y

41

Example 1.4.14

3 3 3

2

33 3 3

2 2

6. 27 3

3 3 9

7. 64 4

4 4 16

b b

b b b

x y x y

x y x xy y

3 3 2 2x y x y x xy y

3 3 2 2x y x y x xy y

42

Example 1.4.14

2

2

2 2

8. 7 12 4 3

9. 42

7 6

10. 4 2 4

w w w w

x y x y

x y x y

x x y

43

1.4 polynomials

Engineering