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