Specialist Maths

Polynomials

Week 1

Definitions

rm.in that te variable theofpower theis terma of degree The

.polynomial in thepower highest theis polynomial theof degree The

power.highest with the variable with the term theis termleading The

valiable.a offront in constant theist coefficienA

powers. integral positive have variables thehererelation w a is polynomialA

0 where

Quartic

Cubic

:Quadratic

:Linear

:Examples

234

23

2

a

edxcxbxax

dcxbxax

cbxax

bax

Addition and subtraction of polynomials (Ex 3A)

104

64243

32243

Example

23

223

223

xxx

xxxx

xxxx

124

3243

3243

Example

23

223

223

xxx

xxxx

xxxx

Multiplication (Ex3A)

3243

Example223 xxxx

Multiplication Solution

3243

Example223 xxxx

Synthetic Multiplication (Ex 3A)

3243

Example223 xxxx

Synthetic Multiplication Solution

3243

Example223 xxxx

Division of Polynomials

algorithm

division thecalled is this

)()()()(

)(

)()(

)(

)(

xRxQxdxP

xd

xRxQ

xd

xP

remainder)(

quotient)(

divisor)(

polynomial)(

where

xR

xQ

xd

xP

5

46

5

46

5

34

Example 1(Ex 3B1)

3

3323

x

xxx

Solution 1

3

3323

x

xxx

Example 2 (Ex 3B1)

2

12 24

x

xxx

Solution 2

2

12 24

x

xxx

Example 3 (Ex 3B2)

1

142

34

xx

xxx

Solution 3

1

142

34

xx

xxx

Synthetic Division (Ex 3B3)

2

12

Example24

x

xxx

Synthetic Division Solution

2

12

Example24

x

xxx

Example 4 (Ex 3B3)

3

122 34

x

xxx

Solution 4

3

122 34

x

xxx

Example5 (Ex 3B3)

12

532 34

x

xxx

Solution 5

12

532 34

x

xxx

Roots, Zeros and Factors

).( of zero a is then ,0)( If xPP

).( ofroot a is then solution, a

is If .0)( solve When we

xP

xxP

)()()(

such that )(Q polynomial a exists

e then ther),( offactor a is )( If

xQxxP

x

xPx

Example 6 (Ex 3C)

xxx 52 of zeros theall Find 23

Solution 6xxx 52 of zeros theall Find 23

Example 7 (Ex 3C)

102

of factorslinear the theall Find2 xx

Solution 7

102

of factorslinear the theall Find2 xx

Example 8 (Ex 3C)

i23 and,3

2 rootswith

polynomial cubic real all Find

Solution 8

i23 and,3

2 rootswith

polynomial cubic real all Find

Example 9 (Ex 3C)

ii 2 and,21 rootswith

spolynomial quartic real all Find

Solution 9

ii 2 and,21 rootswith

spolynomial quartic real all Find

This Week

• Text P80 – 89

• Ex3A Q1-3;

• Ex3B1Q1-3;

• Ex3B2 Q1-3;

• Ex3B3 Q1,2;

• Ex3C Q1-6