specialist mathematics polynomials week 3. graphs of cubic polynomials
TRANSCRIPT
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Specialist Mathematics
PolynomialsWeek 3
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Graphs of Cubic Polynomials
3 xay
2 xxay
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xxxay
04 where 2
2
acb
cbxaxxay
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Graphs of Quartic Polynomials 4 xay 3 xxay
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xxxay 2
xxxxay
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04 where 2
2
acb
cbxaxxxy
04 and ,04 where 22
22
dfeacb
fexdxcbxaxy
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Example 20 (Ex 3F1)
graph. following in theshown cubic theofequation theFind
2 3
36
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Solution 20
graph. following in theshown cubic theofequation theFind
2 3
36
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Example 21(Ex 3F2)
graph. following in theshown quartic theofequation theFind
1
6
3
(-1,24)
(2,-6)
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Solution 21graph. following
in theshown quartic theofequation theFind
1
6
3
(-1,24)
(2,-6)
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Example 22 (Ex 3F2)
(1,4) through passes and 3at axisy thecuts 1,-at it touches
,21at axis x thecuts that quartic theofequation theFind
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Solution 22
(1,4) through passes and 3at axisy thecuts 1,-at it touches
,21at axis x thecuts that quartic theofequation theFind
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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.
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Example 23 (Ex 3G1)
form. surdsimplest in 3732 of roots theall Find 23 xxx
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Solution 23
form. surdsimplest in 3732 of roots theall Find 23 xxx
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Example 24 (Ex 3G1)
form. surdsimplest in 24723 of roots theall Find 234 xxxx
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Solution 24
form. surdsimplest in 24723 of roots theall Find 234 xxxx
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Example 25 (Ex 3G2)
cubic. theof zeros theall hence and find real, is where
,309)( of zero a is 3 If 23
aaaxxaxxPi
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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
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Example 26 (Ex 3G2)
zeros. theall and find real, is Given imaginary.purely is 15101 of zero One 23
aaxxaax
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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
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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