specialist maths polynomials week 1. definitions

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Specialist Maths Polynomials Week 1

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Page 1: Specialist Maths Polynomials Week 1. Definitions

Specialist Maths

Polynomials

Week 1

Page 2: Specialist Maths Polynomials Week 1. Definitions

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

Page 3: Specialist Maths Polynomials Week 1. Definitions

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

Page 4: Specialist Maths Polynomials Week 1. Definitions

Multiplication (Ex3A)

3243

Example223 xxxx

Page 5: Specialist Maths Polynomials Week 1. Definitions

Multiplication Solution

3243

Example223 xxxx

Page 6: Specialist Maths Polynomials Week 1. Definitions

Synthetic Multiplication (Ex 3A)

3243

Example223 xxxx

Page 7: Specialist Maths Polynomials Week 1. Definitions

Synthetic Multiplication Solution

3243

Example223 xxxx

Page 8: Specialist Maths Polynomials Week 1. Definitions

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

Page 9: Specialist Maths Polynomials Week 1. Definitions

Example 1(Ex 3B1)

3

3323

x

xxx

Page 10: Specialist Maths Polynomials Week 1. Definitions

Solution 1

3

3323

x

xxx

Page 11: Specialist Maths Polynomials Week 1. Definitions

Example 2 (Ex 3B1)

2

12 24

x

xxx

Page 12: Specialist Maths Polynomials Week 1. Definitions

Solution 2

2

12 24

x

xxx

Page 13: Specialist Maths Polynomials Week 1. Definitions

Example 3 (Ex 3B2)

1

142

34

xx

xxx

Page 14: Specialist Maths Polynomials Week 1. Definitions

Solution 3

1

142

34

xx

xxx

Page 15: Specialist Maths Polynomials Week 1. Definitions

Synthetic Division (Ex 3B3)

2

12

Example24

x

xxx

Page 16: Specialist Maths Polynomials Week 1. Definitions

Synthetic Division Solution

2

12

Example24

x

xxx

Page 17: Specialist Maths Polynomials Week 1. Definitions

Example 4 (Ex 3B3)

3

122 34

x

xxx

Page 18: Specialist Maths Polynomials Week 1. Definitions

Solution 4

3

122 34

x

xxx

Page 19: Specialist Maths Polynomials Week 1. Definitions

Example5 (Ex 3B3)

12

532 34

x

xxx

Page 20: Specialist Maths Polynomials Week 1. Definitions

Solution 5

12

532 34

x

xxx

Page 21: Specialist Maths Polynomials Week 1. Definitions

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

Page 22: Specialist Maths Polynomials Week 1. Definitions

Example 6 (Ex 3C)

xxx 52 of zeros theall Find 23

Page 23: Specialist Maths Polynomials Week 1. Definitions

Solution 6xxx 52 of zeros theall Find 23

Page 24: Specialist Maths Polynomials Week 1. Definitions

Example 7 (Ex 3C)

102

of factorslinear the theall Find2 xx

Page 25: Specialist Maths Polynomials Week 1. Definitions

Solution 7

102

of factorslinear the theall Find2 xx

Page 26: Specialist Maths Polynomials Week 1. Definitions

Example 8 (Ex 3C)

i23 and,3

2 rootswith

polynomial cubic real all Find

Page 27: Specialist Maths Polynomials Week 1. Definitions

Solution 8

i23 and,3

2 rootswith

polynomial cubic real all Find

Page 28: Specialist Maths Polynomials Week 1. Definitions

Example 9 (Ex 3C)

ii 2 and,21 rootswith

spolynomial quartic real all Find

Page 29: Specialist Maths Polynomials Week 1. Definitions

Solution 9

ii 2 and,21 rootswith

spolynomial quartic real all Find

Page 30: Specialist Maths Polynomials Week 1. Definitions

This Week

• Text P80 – 89

• Ex3A Q1-3;

• Ex3B1Q1-3;

• Ex3B2 Q1-3;

• Ex3B3 Q1,2;

• Ex3C Q1-6