Chapter1IntroPolynomials(S)

8/10/2019 Chapter1IntroPolynomials(S)

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POLYNOMIALS

1. 1 Introduction to Polynomials

1. 2 Remainder Theorem

1.6 Partial Fractions

1.5 Binomial Theorem

1.3 Factor Theorem1.4 Binomial Expansion


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(c) To perform addition, subtraction and

multiplication of polynomials.

(d) To perform division of polynomials and write

the answer in the form P(x)=Q(x)D(x)+R(x),

where the divisor can be linear or quadratic.


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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


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DEFINITION

A polynomial P(x) of degree n is defined as1

1 1 0( ) ... ; 0n n

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

where

Znand

n210 aaaa ,...,,

are called the coefficient of the polynomial


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(i) The coefficient of the highest power ofx,

, is the leading coefficientan

(ii) The constant term is a0

NOTE THAT:

(iii) The degree of the polynomial is

determined by the highest power of x


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Examples of polynomial functions:

Polynomial Degree Name

using

Degree

Leading

coefficient

Constant

term

5 0 Constant

7x + 4 1 Linear

3x2+ 2x + 1 2 Quadratic

4x3-1 3 Cubic

9x4+ 11x+5 4 quartic

0

713

4 -19 5

54


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Examples of non-polynomial expressions:

,x4x 31

,1

x3

x

5

3x3x 2

3xx 31 containsnon-positive

power of x.


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MONOMIALS, BINOMIALS AND TRINOMIALS

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

trinomials, respectively.

Name Example

Monomial

BinomialTrinomial

3x

xx 2

353 23 xx


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ALGEBRAIC

OPERATIONS

+

OPERATIONS OF POLYNOMIAL


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ADDITION AND SUBTRACTION

The addition and subtraction of the polynomial

can be performed by collecting similarterms.

)(xP )(xQand


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EXAMPLE

Given and

Determine

(a)

(b)

452)(34

xxxP.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 multipliedby each term in the other polynomial.

MULTIPLICATION


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Given and

Determine

(a)

(b)

1)(2

xxxP.12)( 23 xxxQ

4Q(x)

P(x)Q(x)

EXAMPLE


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

448 23 xx

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

1232 2345 xxxxx

SOLUTION


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IfP(x) is a polynomial

of degreem

and

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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Dividing Polynomials

Objectives:

To divide a polynomial by a monomial

To divide a polynomial by a polynomial


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Divide by 2.2

8x 4x 1

28x 4x 1

2

28x 4x 1

2 2 2

2 14x 2x2

Example


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Practice!!!

1)

Divide.

34x 6x 5

2

2)3 2

2x 6x 4x

2x


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Divide by .2

x 3x 2 x 1

2x 1 x 3x 2

x

x2 + x- ( )

2x + 2

+ 2

2x + 2- ( )

0

CHECK: (x + 1)(x + 2)

= x2 + 2x + x + 2

= x2 + 3x + 2

Example


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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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LONG

DIVISION

2

35 17

1

34

352 17

2

1

Remainder

DivisorQuotient

1)2)(17(35 Hence,


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Divide by

EXAMPLE

1x632 2 xx


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12

7

1

622

63212

2

x

x

xxx

xxx

1

712

1

632 2

x

x

x

xx

7)1)(12(632 2 xxxx

SOLUTION


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Determine

by using long division.

EXAMPLE

43

743 23

x

xxx


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Determine

EXAMPLE

31634

xx

xxx


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Practice !!

i) Divide by

ii) Divide by

6792 23 xxx 12 x

6512x: 2 xxAnswer

152134 23 xxx 54 x

3254x: 2 xxAnswer

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Chapter1IntroPolynomials(S)