Specialist Mathematics

PolynomialsWeek 3

Graphs of Cubic Polynomials

3 xay

2 xxay

xxxay

04 where 2

2

acb

cbxaxxay

Graphs of Quartic Polynomials 4 xay 3 xxay

xxxay 2

xxxxay

04 where 2

2

acb

cbxaxxxy

04 and ,04 where 22

22

dfeacb

fexdxcbxaxy

Example 20 (Ex 3F1)

graph. following in theshown cubic theofequation theFind

2 3

36

Solution 20

graph. following in theshown cubic theofequation theFind

2 3

36

Example 21(Ex 3F2)

graph. following in theshown quartic theofequation theFind

1

6

3

(-1,24)

(2,-6)

Solution 21graph. following

in theshown quartic theofequation theFind

1

6

3

(-1,24)

(2,-6)

Example 22 (Ex 3F2)

(1,4) through passes and 3at axisy thecuts 1,-at it touches

,21at axis x thecuts that quartic theofequation theFind

Solution 22

(1,4) through passes and 3at axisy thecuts 1,-at it touches

,21at axis x thecuts that quartic theofequation theFind

Polynomial Theorems• Every real polynomial of degree n can be

factorized into n real or complex linear factors.• Complex roots of real polynomials come in

conjugate pairs.• Every real polynomial can be factorized into

combinations of real linear and real quadratic factors.

• Every real polynomial of odd degree has at least one real linear factor.

Example 23 (Ex 3G1)

form. surdsimplest in 3732 of roots theall Find 23 xxx

Solution 23

form. surdsimplest in 3732 of roots theall Find 23 xxx

Example 24 (Ex 3G1)

form. surdsimplest in 24723 of roots theall Find 234 xxxx

Solution 24

form. surdsimplest in 24723 of roots theall Find 234 xxxx

Example 25 (Ex 3G2)

cubic. theof zeros theall hence and find real, is where

,309)( of zero a is 3 If 23

aaaxxaxxPi

Solution 25

23or 3 when 0

32106

True 2 221018 126

1018 96330)1018()63(

30101863

3106

offactor quadratic a is 106

10933Product

633Sumroot. a is 3 thereforeroot, a is 3

309

2

23

223

2

2

2

23

xixxP

xxxxP

aa

aaaxaxaax

axxaxxax

axxxxP

xPxx

iii

iiii

axxaxxP

Example 26 (Ex 3G2)

zeros. theall and find real, is Given imaginary.purely is 15101 of zero One 23

aaxxaax

Solution 26

5 ,53

2 105

101015 23 x,5 11015 10

0 15 10 1

2

325

isfactor quadraticA

Product

0Sumroot. a

also is therefore, beroot imaginary purely theisLet 15101Let

2

2

22

2223

222

22

222

23

m, mb

aa

aa

xaaa

m

xPbmamba

abmxambxx

xxxPbaxmxxP

mx

mimmimi

mimi

mimixxaaxxP

This Week• Text Book Pages 97 to 110• Page 99 Ex 3F1 Q1 – 5• Page 102 Ex 3F2 Q1 – 4• Page 105 Ex 3G1 Q1 – 4• Page 107 Ex 3G2 Q1 - 6