Maths Extension 1 Year 11 - Module 2 - Lesson 8
Further Sums and Products, Symmetric Identities
Name
Date
Class
1. Lesson focus
Symmetric identities in roots. Use α²+β²=(α+β)²−2αβ and α³+β³=(α+β)³−3αβ(α+β) to reduce symmetric expressions.
Success criteria
- Apply the lesson method accurately in a routine example.
- Show each algebraic or domain step and check the result.
- Explain why the method and restrictions fit the question.
2. Worked practice
2. For the same α,β, find α³+β³.
3. Find (α−β)² when α+β=5 and αβ=6.
3. Challenge
If α+β=4 and αβ=−3, find α²+β² and α³+β³.
4. Multiple choice
If α+β=s and αβ=p, α²+β² equals what?
A. s+p
B. s²+2p
C. p²−2s
D. s²−2p
Worked answer key
Compare each step with your working. Keep stated intervals, domains and restrictions in your final answer.
1. If α+β=5 and αβ=6, find α²+β².
25−12=13.
2. For the same α,β, find α³+β³.
5³−3·6·5=125−90=35.
3. Find (α−β)² when α+β=5 and αβ=6.
(α−β)²=(α+β)²−4αβ=25−24=1.
Challenge. If α+β=4 and αβ=−3, find α²+β² and α³+β³.
α²+β²=16+6=22; α³+β³=64−3(−3)(4)=100.
Multiple choice. If α+β=s and αβ=p, α²+β² equals what?
Correct answer: D. Expand (α+β)²=α²+2αβ+β² and subtract 2p.
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