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Introduction

Basic functions:

( ) (Constant function)f x b

( ) , a 0 (Linear function)f x ax b 2( ) , a 0 (Quadratic function)f x ax bx c

3 2( ) , a 0, (Cubic function)f x ax bx cx d

All these functions are special cases of the

general class of functions called

Polynomial Functions

3. 1 POLYNOMIALS

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Definition

A polynomial P(x) of degreen is defined as1

1 1 0( ) ... ; 0

n n

n n nP x a x a x a x a a

where

Znand

n210aaaa ,...,,

called the coefficient of the polynomial.

are

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(i) The coefficient of the highest power ofx,

,is the leading coefficient .an

(ii) The constant term is a0 .

Note that:

(iii) The degree of the polynomial is

determined by the highest power ofx.

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Examples of polynomial functions:

Polynomials Deg NameLeading

coefft

Const

term

7xP )(

65)( xxPxxxP 27)(

23x7x2xP )(

3x 55)(

4 xxxP

0

12

3

4

0

57

2

1

0

7

6

3

5

const.

linearquadratic

cubic

quartic

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Examples of non-polynomial expressions:

,x4x 31

,1x3

x

5

3x3x2

3xx 31 containsnon-positive

power ofx.

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Monomials, Binomials And Trinomials

Polynomials with one, two and three termsare called monomials, binomials and

trinomials, respectively.

Example Name Example

Monomial

BinomialTrinomial

3x

xx 233

127 23 xx

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Algebraic Operations

+

Laws ofNumbers

commutative associative distributive

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Addition and subtraction

The addition and subtraction of the polynomial

can be performed by collecting like terms.)(xP )(xQand

(similar terms)

43Q 234)(

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Example

Given and

Determine

(a)

(b)

452)(34

xxxP

x4x3xxxQ 234 )(

.43)( 234 xxxxxQ

P(x) + Q(x)

P(x) Q(x)

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Solution

)43452 23434 xxx(x)xx(

44343 234 xxxx

)()((a) xQxP

)43()452( 23434 xxxxxx

)()()b( xQxP

4436 234 xxxx

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Every term in one polynomial is multiplied

by each term in the other polynomial.

Multiplication

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Example

Given and

Determine

(a)

(b)

1)(2

xxxP.12)( 23 xxxQ

4Q(x)

P(x)Q(x)

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Solution

)12(4)(4)a( 23 xxxQ

448 23 xx

)12)(1()()()b( 232 xxxxxQxP

1232 2345 xxxxx

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IfP(x) is a polynomialof degree mand

Q(x) is a polynomial

of degree n,Then

productP(x)Q(x) isa polynomial of

degree (m + n)

Note that:

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Division

The division of the polynomial can be

expressed in the form

)()()()( xRxQxDxP where

R(x) Remainder

D(x) Divisor Q(x) Quotient

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Solution 12

7

1

6

22

6321

2

2

x

x

x

xx

xxx

1

712

1

632 2

x

x

x

xx

7)1)(12(632 2 xxxx

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Example

Determine

by using long division.

43

743 23

x

xxx

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Solution 1

112

43

743

74343

2

23

23

x

x

x

xxx

xxxx

43

211)1(

43

743 223

x

xx

x

xxx

)211()43)(1(743 223 xxxxxx

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Example

Divide by42 267 xx 52

x

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Solution 162

87

8016

716

102

7625

2

2

2

24

242

x

x

x

xx

xxx

5

87)162(

5

7622

2

2

24

xx

x

xx

87)5)(162(762 2224 xxxx

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Example

Determine

)3)(1(

634

xx

xxx

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93

2126

27369-)

6109

9123

63334

6034

2

2

2

22

23

234

2342

xx

x

xx

xx

xxx

xxxxxx

xxxxxxSolution

)2126()3)(1)(93(6 234 xxxxxxxx