Chapter 1.4, Part 1 The Ring of Polynomials.pdf

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Chapter 1.4Chapter 1.4THE RING OF THE RING OF

POLYNOMIALSPOLYNOMIALS

Types of QuantitiesTypes of Quantities

Variables represented only by lettersand whose values may bearbitrarily chosen dependingon the situation.

Types of QuantitiesTypes of Quantities

Constanta quantity whose value is fixedand may not be changed duringa particular discussion.

Example 1.4.1Example 1.4.1

Consider the formula - force - mass

- acceleration

, , and are variables.

F maFma

F m a


In the formula for computing the circumference of a circle

and are constants,

while is a varia

2

2

ble.

r

r

Algebraic ExpressionsAlgebraic Expressions

Any combination of numbers andsymbols related by the operationsfrom the previous sections will be

algebraic excalle pressd an ion.Addition, subtraction, multiplication, division, powers and roots

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2 2

3

2 3

2

2are algebraic expressions

x yxx y xy xy

x x


2

Any algebraic expression consistingof distinct parts sepa

algerated by

braic sum+ or

is called an .

2 3 2 5x xy xy


2

Each distinct part, together with itssign, is called a of the algebraicex

tp

ermression.

2 3 2 5x xy xy

Types of Algebraic Types of Algebraic ExpressionsExpressions

An algebraic expression consisting of1 term - monomial2 terms - binomial3 terms - trinomialany number of terms - multinomial


Each factor of a term may be callcoefficient

edthe 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

tliteral coefficient.

u vu v

u vv u

u v


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2 2

T e r m s h a v in g t h e s a m e l i t e r a l c o e f f i c i e n t s i m i l a rt e r m s

s a r e c a l l e d .

3 2 5x x y x y

PolynomialsPolynomials

A is an algebraicexpression involving only non-negative integra

polynomi

l powers

al in

of .

t

t


2 3Is 2 3 1 a polynomial in ?

Is it a polynomial in ?

Is it a polynomial in ?

xt yt t

x

y


2

2 2

A polynomial in can be expressedas .

2 1A polynomial in and can beexpressed , as .

,

P t

Q

t

P t t tx y

Q x y xx

yy


The notations can be used tofind the value of the expressionwhen the value of the variableis given.

1If , then2

14 4 2210 0 02

S t gt

S g g

S g

If 7, then1 7

7 is constant polynomia al .

C xCC

C

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The of a term of a polynomial is the exponent of the

variable.

The of a

degreein a variable

degree insome variab

term of a polynomial is the sum of the degrees

of all variables in that tles

erm.


2

3 4

4 10

The degree of in is 2.

The degree of in is 4.

The degree of in and is 14.

x y x

x y y

x y x y


The degree of the polynomial in somevariables is that of its term of degree in those vari

highestables.


3

4 3 2 2 3

1. 3 2The degree of in is 3.

2. , 3 2The degree of in is 4.The degree of in is 3.The degree of in and is 4.

P x x xP x

Q x y x x y x y yQ xQ yQ x y

Addition of PolynomialsAddition 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 xQ x x x x

P x Q x x x x

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Multiplication of PolynomialsMultiplication of Polynomials

distributive propertyTo multiply two polynomials, weapply the ,the , and laws of exponents add similar terms.


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 15 106 15 4 10

t t t t t

t t t

t t tt t t


2 2

3 4 2 2 2 2

3 4 2 2 2

2. 2 3 2 4

6 4 8 3 2 46 4 10 3 4

x y xy x y

x y x x y xy x y yx y x x y xy y

Special ProductsSpecial Products

2 2

Binomial Sum and Differencex y x y x y


2

3 3 2 6

2 3 2 3 4 2 6

Perform the indicated operation.Use special products.

1. 2 2 42. 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

Chapter 1.4, Part 1 The Ring of Polynomials.pdf

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Transcript of Chapter 1.4, Part 1 The Ring of Polynomials.pdf