3.2

Polynomials

Types of Quantities

represented only by letters

and whose values may be

arbitrarily chosen depending

on the si

Varia

tuat

bles

ion.

Types of Quantities

a quantity whose value is fixed

and may not be changed during

a particular disc

Cons

uss

tant

ion.

Example 1.4.1

- force

- mass

- acceleration

, , and are variables.

F ma

F

m

a

F m a

Example 1.4.1

21

2

is a variable.

1 is a constant.

2

is a constant referring to the

value of gravity.

gt

t

g

Algebraic Expressions

Any combination of numbers and

symbols related by the operations

from the previous sections will be

algebraic excalle pressd an ion.

Example 1.4.2

2 2

3 2

2 3

2

are algebraic expressions

x y

xx y xy x

y

x x

Algebraic Sum

2

Any algebraic expression consisting

of distinct parts sepa

alge

rated by

braic sum

+ or

is called an .

2 3 2 5x xy xy

Term

Coefficient

Each factor of a term may be call

coefficient

ed

the of the others.

Example 1.4.3

4

4

numerical coefficie

In 3 ,

3 is the .

is

nt

literal coefficientthe .

u v

u v

Similar Terms

2 2

Terms having the same literal

coefficient similar

terms

s are called

.

3 2 5x xy x y

Polynomials

A is an algebraic

expression involving only non-

negative integral powers of one

or more variable and containing

no variable in the de

polynomi

nomina

al

tor.

Which of the following is a polynomial?

2 21) 3x y

32) 3x y

3)

x y

x y

2 34) x y

3 25) 2x x

2 4

6)3 7

x y

Polynomials

Degree of a Term

The of a term of a polynomial

is the sum of the powers of all variables

in the

degree

term.

Example 1.4.6

5

4

3 3

2

7

Find the degree of each term.

2

3

12

x

x y

x y

x

y

Degree of a Polynomial

The degree of the polynomial

is that of its term of highest

degree.

Example 1.4.7

3

4 3 2 2 3

1. 3 2

degree 3

2. 3 2

degree 4

x x

x x y x y y

-According to degrees

CONSTANT degree 0

LINEAR degree 1

QUADRATIC degree 2

CUBIC degree 3

7

x y z

xy x

2 33 3 x x x

Types of Polynomials

Addition of Polynomials

To add polynomials, w add

similar t

e

erms.

Example 1.4.8

3 2 3 2

3 2

3 2

3 2

Add 2 1 and 3 5 10.

2 1

3 5 10

2 5 9

x x x x x

x x

x x x

x x x

Example: Subtract the 2nd polynomial

from the 1st one.

3 2

3 2

4 7 2 4

3 8 3 7

x x x

x x x3x 215x x 3

Subtraction of Polynomials

Multiplication of Polynomials

distributive property

To multiply two polynomials, we

apply the ,

the , and laws of exponents add

similar terms.

Example 1.4.9

2 2 2

3 2

3 2

3 2

Perform the indicated operations.

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

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

2 23 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

Let and be polynomials

and be a non-zero mononial.

Then

.

P x R x

Q x

P x R x P x R x

Q x Q x Q x

Division of Polynomials by Monomial

6 3 6 3

2 2 2 2

4

5 2 3 5 2 3

4 2

20 5 2 20 5 21.

2 2 2 2

5 110

2

12 60 9 12 60 92.

3 3 3 3

34 20

x x x x x x

x x x x

x xx

x y x y x y x y

xy xy xy xy

x y xxy

Example

General Division of Polynomials

RemainderQuotient +

where and are polynomials

such that 0 and the degree of

is less than or equal to the degree

of .

P x

Q x Q x

P x Q x

Q x

Q x

P x

2

2

2

Divide 5 6 by 1.

126

1

By long division: 1 5 6

6 6

6 6

12

x x x

xx

x x x

x x

x

x

SYNTHETIC DIVISION

2 5 6

1

x x

x

1. Write the numerical coefficients

(of P(x) in descending power) with a

at the left side: 1 5 61

1

Form:

P x

x a

1

6

6

12

Quotient= 6x

Remainder= 12

2. Bring down the leading coefficient:

3. Multiply this brought-down value by

a, and carry the result up into the

next column:

4. Add down the column:

5. Do Steps 3 and 4 repeatedly up

to the last column: 2 5 6 12

61 1

x xx

x x

2 3

2 3 3 2

2 3 6Solve using Synthetic Division.

2

2 3 6 3 2 6 ; 2 2

x x

x

x x x x x x

-3 -2 0 62

-3

6

4

-8

-8

2Quotient = 3 4 8x x

Remainder = 22

16

22

3 223 2 6 22 -3x 4 8

2 2

x xx

x x

Removing Grouping Symbols

Remove grouping symbols and simplify.

2

2

2

1. 2 4 5 1

2 4 5

2 20 4 4

6 20 4

x x x

x x x

x x x

x x

Removing Grouping Symbols

2. 4 3 2 6 5

4 3 12 2 5

4 17

4 17

21

x x

x x

x

x

x

Recitation Activity

Integer Exponents (3.1)

Operations on Polynomials (3.2)

End of Chapter 3.2